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lived there so he could obtain permission to study at the Charles University without having to find a job to support himself. One should also not underestimate the fact that by this stage his mathematical preferences were very much in place and Karl Loewner in Prague looked the ideal supervisor. Indeed Bers did obtain his doctorate which was awarded in |
1938 from the Charles University of Prague where he wrote a thesis on potential theory under Karl Loewner's supervision. At the time Bers was rather unhappy with Loewner :- Lipa spoke of feeling neglected, perhaps even not encouraged, by Loewner and said that only in retrospect did he understand Loewner's teaching method. He gave to each of his students the |
amount of support needed ... It is obvious that Lipa did not appear too needy to Loewner. In 1938 Czechoslovakia became an impossible country for someone of Jewish background. Equally dangerous was the fact that Bers had no homeland since he was a wanted man in Latvia, and was a left wing academic. With little choice but to escape again, |
Bers fled to Paris where his daughter Ruth was born. However, the war followed him and soon the Nazi armies began occupying France. Bers applied for a visa to the USA and, while waiting to obtain permission, he wrote two papers on Green's functions and integral representations. Just days before Paris surrendered to the advancing armies, Bers and his family |
moved from Paris to a part of France not yet under attack from the advancing German armies. At last he received the news that he was waiting for, the issue of American visas for his family. In 1940 Bers and his family arrived in the United States and joined his mother who was already in New York. There was of |
course a flood of well qualified academics arriving in the United States fleeing from the Nazis and there was a great scarcity of posts, even for the most brilliant, so he was unemployed until 1942, living with other unemployed refugees in New York. During this time he continued his mathematical researches. After this he was appointed Research Instructor at Brown |
University where, as part of work relevant to the war effort, he studied two-dimensional subsonic fluid flow. This was important at that time since aircraft wings were being designed for planes with jet engines capable of high speeds. Between 1945 and 1949 Bers worked at Syracuse University, first at Assistant Professor, later as Associate Professor. Gelbart wanted to build up |
the department at Syracuse and attracting both Bers and Loewner was an excellent move. Here Bers began work on the problem of removability of singularities of non-linear elliptic equations. His major results in this area were announced by him at the International Congress of Mathematicians in 1950 and his paper Isolated singularities of minimal surfaces was published in the Annals |
of Mathematics in 1951. Courant writes:- The nonparametric differential equation of minimal surfaces may be considered the most accessible significant example revealing typical qualities of solutions of non-linear partial differential equations. With a view to such a general objective, [Bers] has studied singularities, branch-points and behaviour in the large of minimal surfaces. Abikoff writes in that this paper is:- ... |
a magnificent synthesis of complex analytic techniques which relate the different parameterisations of minimal surfaces to the representations of the potential function for subsonic flow and thereby achieves the extension across the singularity. Bers then became a member of the Institute for Advanced Study at Princeton where he began work on Teichmüller theory, pseudoanalytic functions, quasiconformal mappings and Kleinian groups. |
He was set in the right direction by an inequality he found in a paper of Lavrentev who attributed the inequality to Ahlfors. In a lecture he gave in 1986 Bers explained what happened next:- I was in Princeton at the time. Ahlfors came to Princeton and announced a talk on quasiconformal mappings. He spoke at the University so I |
went there and sure enough, he proved this theorem. So I came up to him after the talk and asked him "Where did you publish it?", and he said "I didn't". "So why did Lavrentev credit you with it?" Ahlfors said "He probably thought I must know it and was too lazy to look it up in the literature". When |
Bers met Lavrentev three years later he asked him the same questions and, indeed, Ahlfors had been correct in guessing why Lavrentev had credited him. Bers continued in his 1986 lecture:- I immediately decided that, first of all, if quasiconformal mappings lead to such powerful and beautiful results and, secondly, if it is done in this gentlemanly spirit - where |
you don't fight over priority - this is something that I should spend the rest of my life studying. It is ironic, given Bers strong political views on human rights, that he should find that Teichmüller, a fervent Nazi, had already made stunning contributions. In one of his papers on Teichmüller theory, Bers quotes Plutarch:- It does not of necessity |
follow that, if the work delights you with its grace, the one who wrought it is worthy of your esteem. In 1951 Bers went to the Courant Institute in New York, where he was a full professor, and remained there for 13 years. During this time he wrote a number of important books and surveys on his work. He published |
Theory of pseudo-analytic functions in 1953 which Protter, in a review, described as follows:- The theory of pseudo-analytic functions was first announced by [Bers] in two notes. These lecture notes not only contain proofs and extensions of the results previously announced but give a self-contained and comprehensive treatment of the subject. The author sets as his goal the development of |
a function theory for solutions of linear, elliptic, second order partial differential equations in two independent variables (or systems of two first-order equations). One of the chief stumbling blocks in such a task is the fact that the notion of derivative is a hereditary property for analytic functions while this is clearly not the case for solutions of general second |
order elliptic equations. Another classic text was Mathematical aspects of subsonic and transonic gas dynamics published in 1958:- It should be said, even though this is taken for granted by everybody in the case of Professor Bers, that the survey is masterly in its elegance and clarity. In 1958 Bers address the International Congress of Mathematicians in Edinburgh, Scotland, where |
he lectured on Spaces of Riemann surfaces and announced a new proof of the measurable Riemann mapping theorem. In his talk Bers summarised recent work on the classical problem of moduli for compact Riemann surfaces and sketched a proof of the Teichmüller theorem characterizing extremal quasiconformal mappings. He showed that the Teichmüller space for surfaces of genus g is a |
(6g-6)-cell, and showed how to construct the natural complex analytic structure for the Teichmüller space. Bers was a Guggenheim Fellow in 1959-60, and a Fulbright Fellow in the same academic year. From 1959 until he left the Courant Institute in 1964, Bers was Chairman of the Graduate Department of Mathematics. In 1964 Bers went to Columbia University where he was |
to remain until he retired in 1984. He was chairman of the department from 1972 to 1975. He was appointed Davies Professor of Mathematics in 1972, becoming Emeritus Davies Professor of Mathematics in 1982. During this period Bers was Visiting Miller Research Professor at the University of California at Berkeley in 1968. Tilla Weinstein describes in Bers as a lecturer:- |
Lipa's courses were irresistible. He laced his lectures with humorous asides and tasty tidbits of mathematical gossip. He presented intricate proofs with impeccable clarity, pausing dramatically at the few most critical steps, giving us a chance to think for ourselves and to worry that he might not know what to do next. Then, just as the silence got uncomfortable, he |
would describe the single most elegant way to complete the argument. Jane Gilman describes Bers' character:- Underneath the force of Bers' personality and vivacity was the force of his mathematics. His mathematics had a clarity and beauty that went beyond the actual results. He had a special gift for conceptualising things and placing them in the larger context. In Bers |
life is summed up by Abikoff as follows:- Lipa possessed a joy of life and an optimism that is difficult to find at this time and that is sorely missed. Those of us who experienced it directly have felt an obligation to pass it on. That, in addition to the beauty of his own work, is Lipa's enduring gift to |
us. We have yet to say something about Bers' great passion for human rights. In fact this was anything but a sideline in his life and one could consider that he devoted himself full-time to both his mathematical work and to his work as a social reformer. Perhaps his views are most clearly expressed by quoting from an address he |
gave in 1984 when awarded an honorary degree by the State University of New York at Stony Brook:- By becoming a human rights activist ... you do take upon yourself certain difficult obligations. ... I believe that only a truly even-handed approach can lead to an honest, morally convincing, and effective human rights policy. A human rights activist who hates |
and fears communism must also care about the human rights of Latin American leftists. A human rights activist who sympathises with the revolutionary movement in Latin America must also be concerned about human rights abuses in Cuba and Nicaragua. A devout Muslim must also care about human rights of the Bahai in Iran and of the small Jewish community in |
Syria, while a Jew devoted to Israel must also worry about the human rights of Palestinian Arabs. And we American citizens must be particularly sensitive to human rights violations for which our government is directly or indirectly responsible, as well as to the human rights violations that occur in our own country, as they do. Bers received many honours for |
his contributions in addition to those we have mentioned above. He was elected to the American Academy of Arts and Sciences, to the Finnish Academy of Sciences, and to the American Philosophical Society. He served the American Mathematical Society in several capacities, particularly as Vice-President (1963-65) and as President (1975-77). The American Mathematical Society awarded him their Steele Prize in |
1975. He received the New York Mayor's award in Science and Technology in 1985. He was an honorary life member of the New York Academy of Sciences, and of the London Mathematical Society. Article by: J J O'Connor and E F Robertson Click on this link to see a list of the Glossary entries for this page List of References |
point| The astroid only acquired its present name in 1836 in a book published in Vienna. It has been known by various names in the literature, even after 1836, including |
cubocycloid and paracycle. The length of the astroid is 6a and its area is 3πa2/8. The gradient of the tangent T from the point with parameter p is -tan(p). The |
equation of this tangent T is x sin(p) + y cos(p) = a sin(2p)/2 Let T cut the x-axis and the y-axis at X and Y respectively. Then the length |
XY is a constant and is equal to a. It can be formed by rolling a circle of radius a/4 on the inside of a circle of radius a. It |
can also be formed as the envelope produced when a line segment is moved with each end on one of a pair of perpendicular axes. It is therefore a glissette. |
The machete blades turned red with heat in the fire that the rubber workers built on a Liberia plantation, Thomas Unnasch remembers from a visit in the 1980s. This was how the men tried to quell the intense itchiness that |
comes with river blindness, a rare tropical disease. "You can imagine how bad the itching must be, that running a red-hot machete up and down your back would be a relief, but it was," said Unnasch, whose laboratory works on |
diagnostic tests for the disease. About 18 million people have river blindness worldwide, according to the World Health Organization, but more than 99% of cases of this disease are found in Africa. It goes by the technical name "onchocerciasis," and |
it spreads through small black flies that breed in fast-flowing, highly oxygenated waters. When an infected fly bites a person, it drops worm larvae in the skin, which can then grow and reproduce in the body. Unlike malaria, river blindness |
is not fatal, but it causes a "miserable life," said Moses Katabarwa, senior epidemiologist for the Atlanta-based Carter Center's River Blindness Program, which has been leading an effort to eliminate the disease in the Americas and several African countries. Some |
strains cause blindness, while others come with more severe skin disease. With time, generally all strains of the disease can lead to rough "lizard" skin, depigmented "leopard skin" and hanging groins. Another big problem among patients is itching, which happens |
when the worms die inside a person. In southwest Uganda, the locals call the disease "Obukamba," referring to the symptoms of distorted skin appearance and itchiness, Katabarwa said. In western Uganda, he said, "the fly is called 'Embwa fly' or |
dog fly, for it bites like a dog!" There is no vaccine for river blindness, but there is a drug, called ivermectin that paralyzes and kills the offspring of adult worms, according to the Mayo Clinic. It may also slow |
the reproduction of adult female worms, so there are fewer of them in the skin, blood and eyes. The pharmaceutical company Merck has been donating the treatment, under the brand name Mectizan, since 1985. Great strides have been made against |
Attention Deficit Hyperactivity Disorder or ADHD is a common childhood illness. People who are affected can have trouble with paying attention, sitting still and controlling their impulses. There are three types of ADHD. The most common type of ADHD is |
when people have difficulties with both attention and hyperactivity. This is called ADHD combined type. Some people only have difficulty with attention and organization. This is ADHD inattentive subtype or Attention Deficit Disorder (ADD). Other people have only the hyperactive |
and impulsive symptoms. This is ADHD hyperactive subtype. It is a health condition involving biologically active substances in the brain. Studies show that ADHD may affect certain areas of the brain that allow us to solve problems, plan ahead, understand |
others' actions, and control our impulses. Many children and adults are easily distracted at times or have trouble finishing tasks. If you suspect that your child has ADHD, it is important to have your child evaluated by his or her |
doctor. In order for your child’s doctor to diagnose your child with ADHD, the behaviors must appear before age 7 and continue for at least six months. The symptoms must also create impairment in at least two areas of the |
child's life-in the classroom, on the playground, at home, in the community, or in social settings. Many children have difficulties with their attention but attention problems are not always cue to ADHD. For example, stressful life events and other childhood |
conditions such as problems with schoolwork caused by a learning disability or anxiety and depression can interfere with attention. According to the National Institute of Mental Health, ADHD occurs in an estimated 3 to 5 percent of preschool and school-age |
children. Therefore, in a class of 25 to 30 children, it is likely that at least one student will have this condition. ADHD begins in childhood, but it often lasts into adulthood. Several studies done in recent years estimate that |
30 to 65 percent of children with ADHD continue to have symptoms into adolescence and adulthood. No one knows exactly what causes ADHD. There appears to be a combination of causes, including genetics and environmental influences Several different factors could |
increase a child's likelihood of having the disorder, such as gender, family history, prenatal risks, environmental toxins and physical differences in the brain seem to be involved. A child with ADHD often shows some of the following: Difficulties with attention: |
- trouble paying attention - inattention to details and makes careless mistakes - easily distracted - losing things such as school supplies - forgetting to turn in homework - trouble finishing class work and homework - trouble listening - trouble |
following multiple adult commands - difficulty playing quietly - inability to stay seated - running or climbing excessively - always "on the go" - talks too much and interrupts or intrudes on others - blurts out answers The good news |
is that effective treatment is available. The first step is to have a careful and thorough evaluation with your child’s primary care doctor or with a qualified mental health professional. With the right treatment, children with ADHD can improve their |
ability to pay attention and control their behavior. The right care can help them grow, learn, and feel better about themselves. Medications: Most children with ADHD benefit from taking medication. Medications do not cure ADHD. Medications can help a child |
control his or her symptoms on the day that the pills are taken. Medications for ADHD are well established and effective. There are two main types: stimulant and non-stimulant medications. Stimulants include methylphenidate, and amphetamine salts. Non-stimulant medications include atomoxetine. |
For more information about the medications used to treat ADHD, please see the Parent Med Guide. Before medication treatment begins, your child's doctor should discuss the benefits and the possible side effects of these medications. Your child’s doctor should continue |
to monitor your child for improvement and side effects. A majority of children who benefit from medication for ADHD will continue to benefit from it as teenagers. In fact, many adults with ADHD also find that medication can be helpful. |
Therapy and Other Support: A psychiatrist or other qualified mental health professional can help a child with ADHD. The psychotherapy should focus on helping parents provide structure and positive reinforcement for good behavior. In addition, individual therapy can help children |
gain a better self-image. The therapist can help the child identify his or her strengths and build on them. Therapy can also help a child with ADHD cope with daily problems, pay better attention, and learn to control aggression. A |
therapist may use one or more of the following approaches: Behavior therapy, Talk therapy, Social skills training, Family support groups. Sometimes children and parents wonder when children can stop taking ADHD medication. If you have questions about stopping ADHD medication, |
consult your doctor. Many children diagnosed with ADHD will continue to have problems with one or more symptoms of this condition later in life. In these cases, ADHD medication can be taken into adulthood to help control their symptoms. For |
others, the symptoms of ADHD lessen over time as they begin to "outgrow" ADHD or learn to compensate for their behavioral symptoms. The symptom most apt to lessen over time is hyperactivity. Some signs that your child may be ready |
to reduce or stop ADHD medication are: - Your child has been symptom-free for more than a year while on medication, - Your child is doing better and better, but the dosage has stayed the same, - Your child's behavior |
is appropriate despite missing a dose or two, - Or your child has developed a newfound ability to concentrate. The choice to stop taking ADHD medication should be discussed with the prescribing doctor, teachers, family members, and your child. You |
may find that your child needs extra support from teachers and family members to reinforce good behavior once the medication is stopped. Without treatment, a child with ADHD may fall behind in school and have trouble with friendships. Family life |
may also suffer. Untreated ADHD can increase strain between parents and children. Parents often blame themselves when they can't communicate with their child. The sense of losing control can be very frustrating. Teenagers with ADHD are at increased risk for |
driving accidents. Adults with untreated ADHD have higher rates of divorce and job loss, compared with the general population. Luckily, safe and effective treatments are available which can help children and adults help control the symptoms of ADHD and prevent |
Weights linked to lower diabetes risk Weight gains Weight training, and not just cardio workouts, is linked to a lower risk of developing type 2 diabetes, according to a US study. "We all know that aerobic exercise is beneficial for |
diabetes - many studies have looked at that - but no studies have looked at weight training," says study leader Frank Hu, at the Harvard School of Public Health. "This study suggests that weight training is important for diabetes, and |
probably as important as aerobic training." Hu and his colleagues, whose report was published in the Archives of Internal Medicine, used data on more than 32,000 male health professionals, who answered questionnaires every two years from 1990 to 2008. On |
average, four out of 1000 men developed type 2 diabetes every year, the researchers found. The risk of getting the blood sugar disorder was only half as high for men who did cardio, or aerobic, workouts - say brisk walking, |
jogging or playing tennis - at least 150 minutes a week, as for those who didn't do any cardio exercise. Men who did weight training for 150 minutes or more had a risk reduction of a third compared to those |
who never lifted weights, independently of whether or not they did aerobic exercise. Exercise is beneficial Whereas weight training increases muscle mass and can reduce abdominal obesity, it tends not to cut overall body mass, says Hu. The results don't |
prove that working out staves off diabetes, because many men who stay fit may also be healthier in other ways, but the researchers did their best to account for such potential differences, including age, smoking and diet. "I think the |
benefits of weight training are real," says Hu. "Any type of exercise is beneficial for diabetes prevention, but weight training can be incorporated with aerobic exercise to get the best results." Along with an appropriate diet, exercise is also important |
Arctic meltdown not caused by nature Rapid loss of Arctic sea ice - 80 per cent has disappeared since 1980 - is not caused by natural cycles such as changes in the Earth's orbit around the Sun, says Dr Karl. |
The situation is getting rather messy with regard to the ice melting in the Arctic. Now the volume of the ice varies throughout the year, rising to its peak after midwinter, and falling to its minimum after midsummer, usually in |
the month of September. Over most of the last 1,400 years, the volume of ice remaining each September has stayed pretty constant. But since 1980, we have lost 80 per cent of that ice. Now one thing to appreciate is |
that over the last 4.7 billion years, there have been many natural cycles in the climate — both heating and cooling. What's happening today in the Arctic is not a cycle caused by nature, but something that we humans did |
by burning fossil fuels and dumping slightly over one trillion tonnes of carbon into the atmosphere over the last century. So what are these natural cycles? There are many many of them, but let's just look at the Milankovitch cycles. |
These cycles relate to the Earth and its orbit around the Sun. There are three main Milankovitch cycles. They each affect how much solar radiation lands on the Earth, and whether it lands on ice, land or water, and when |
it lands. The first Milankovitch cycle is that the orbit of the Earth changes from mostly circular to slightly elliptical. It does this on a predominantly 100,000-year cycle. When the Earth is close to the Sun it receives more heat |
energy, and when it is further away it gets less. At the moment the orbit of the Earth is about halfway between "nearly circular" and "slightly elliptical". So the change in the distance to the Sun in each calendar year |
is currently about 5.1 million kilometres, which translates to about 6.8 per cent difference in incoming solar radiation. But when the orbit of the Earth is at its most elliptical, there will be a 23 per cent difference in how |
much solar radiation lands on the Earth. The second Milankovitch cycle affecting the solar radiation landing on our planet is the tilt of the north-south spin axis compared to the plane of the orbit of the Earth around the Sun. |
This tilt rocks gently between 22.1 degrees and 24.5 degrees from the vertical. This cycle has a period of about 41,000 years. At the moment we are roughly halfway in the middle — we're about 23.44 degrees from the vertical |
and heading down to 22.1 degrees. As we head to the minimum around the year 11,800, the trend is that the summers in each hemisphere will get less solar radiation, while the winters will get more, and there will be |
a slight overall cooling. The third Milankovitch cycle that affects how much solar radiation lands on our planet is a little more tricky to understand. It's called 'precession'. As our Earth orbits the Sun, the north-south spin axis does more |
than just rock gently between 22.1 degrees and 24.5 degrees. It also — very slowly, just like a giant spinning top — sweeps out a complete 360 degrees circle, and it takes about 26,000 years to do this. So on |
January 4, when the Earth is at its closest to the Sun, it's the South Pole (yep, the Antarctic) that points towards the Sun. So at the moment, everything else being equal, it's the southern hemisphere that has a warmer |
summer because it's getting more solar radiation, but six months later it will have a colder winter. And correspondingly, the northern hemisphere will have a warmer winter and a cooler summer. But of course, "everything else" is not equal. There's |
more land in the northern hemisphere but more ocean in a southern hemisphere. The Arctic is ice that is floating on water and surrounded by land. The Antarctic is the opposite — ice that is sitting on land and surrounded |
by water. You begin to see how complicated it all is. We have had, in this current cycle, repeated ice ages on Earth over the last three-million years. During an ice age, the ice can be three kilometres thick and |
cover practically all of Canada. It can spread through most of Siberia and Europe and reach almost to where London is today. Of course, the water to make this ice comes out of the ocean, and so in the past, |
the ocean level has dropped by some 125 metres. From three million years ago to one million years ago, the ice advanced and retreated on a 41,000-year cycle. But from one million years ago until the present, the ice has |
advanced and retreated on a 100,000-year cycle. What we are seeing in the Arctic today — the 80 per cent loss in the volume of the ice since 1980 — is an amazingly huge change in an amazingly short period |
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