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Wikipedia:Neda Bokan#0
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Neda Bokan (born 1947) is a Serbian mathematician specializing in differential geometry. == Education and career == Bokan joined the Mathematical Institute of the Serbian Academy of Sciences and Arts as an assistant in 1969, began working at the University of Belgrade in 1971, and completed a Ph.D. there in 1979, with a dissertation on transformation groups of almost-contact manifolds supervised by Mileva Prvanović. She eventually became full professor at the University of Belgrade, served as dean of mathematics from 1998 to 2001 and again from 2003 to 2007, and was vice rector for education from 2006 to 2012. She also worked as a full professor at the State University of Novi Pazar from 2012 to 2015. She was president of the National Entity for Accreditation and Quality Assurance in Higher Education until stepping down from that position in 2019. She is the author of ten textbooks, and in 2013 became editor-in-chief of the journal Matematički Vesnik, the journal of the Mathematical Society of Serbia. == References ==
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Wikipedia:Negative controls#0
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A scientific control is an experiment or observation designed to minimize the effects of variables other than the independent variable (i.e. confounding variables). This increases the reliability of the results, often through a comparison between control measurements and the other measurements. Scientific controls are a part of the scientific method. == Controlled experiments == Controls eliminate alternate explanations of experimental results, especially experimental errors and experimenter bias. Many controls are specific to the type of experiment being performed, as in the molecular markers used in SDS-PAGE experiments, and may simply have the purpose of ensuring that the equipment is working properly. The selection and use of proper controls to ensure that experimental results are valid (for example, absence of confounding variables) can be very difficult. Control measurements may also be used for other purposes: for example, a measurement of a microphone's background noise in the absence of a signal allows the noise to be subtracted from later measurements of the signal, thus producing a processed signal of higher quality. For example, if a researcher feeds an experimental artificial sweetener to sixty laboratories rats and observes that ten of them subsequently become sick, the underlying cause could be the sweetener itself or something unrelated. Other variables, which may not be readily obvious, may interfere with the experimental design. For instance, the artificial sweetener might be mixed with a dilutant and it might be the dilutant that causes the effect. To control for the effect of the dilutant, the same test is run twice; once with the artificial sweetener in the dilutant, and another done exactly the same way but using the dilutant alone. Now the experiment is controlled for the dilutant and the experimenter can distinguish between sweetener, dilutant, and non-treatment. Controls are most often necessary where a confounding factor cannot easily be separated from the primary treatments. For example, it may be necessary to use a tractor to spread fertilizer where there is no other practicable way to spread fertilizer. The simplest solution is to have a treatment where a tractor is driven over plots without spreading fertilizer and in that way, the effects of tractor traffic are controlled. The simplest types of control are negative and positive controls, and both are found in many different types of experiments. These two controls, when both are successful, are usually sufficient to eliminate most potential confounding variables: it means that the experiment produces a negative result when a negative result is expected, and a positive result when a positive result is expected. Other controls include vehicle controls, sham controls and comparative controls. == Confounding == Confounding is a critical issue in observational studies because it can lead to biased or misleading conclusions about relationships between variables. A confounder is an extraneous variable that is related to both the independent variable (treatment or exposure) and the dependent variable (outcome), potentially distorting the true association. If confounding is not properly accounted for, researchers might incorrectly attribute an effect to the exposure when it is actually due to another factor. This can result in incorrect policy recommendations, ineffective interventions, or flawed scientific understanding. For example, in a study examining the relationship between physical activity and heart disease, failure to control for diet, a potential confounder, could lead to an overestimation or underestimation of the true effect of exercise. Falsification tests are a robustness-checking technique used in observational studies to assess whether observed associations are likely due to confounding, bias, or model misspecification rather than a true causal effect. These tests help validate findings by applying the same analytical approach to a scenario where no effect is expected. If an association still appears where none should exist, it raises concerns that the primary analysis may suffer from confounding or other biases. Negative controls are one type of falsification tests. The need to use negative controls usually arise in observational studies, when the study design can be questioned because of a potential confounding mechanism. A Negative control test can reject study design, but it cannot validate them. Either because there might be another confounding mechanism, or because of low statistical power. Negative controls are increasingly used in the epidemiology literature, but they show promise in social sciences fields such as economics. Negative controls are divided into two main categories: Negative Control Exposures (NCEs) and Negative Control Outcomes (NCOs). Lousdal et al. examined the effect of screening participation on death from breast cancer. They hypothesized that screening participants are healthier than non-participants and, therefore, already at baseline have a lower risk of breast-cancer death. Therefore, they used proxies for better health as negative-control outcomes (NCOs) and proxies for healthier behavior as negative-control exposures (NCEs). Death from causes other than breast cancer was taken as NCO, as it is an outcome of better health, not effected by breast cancer screening. Dental care participation was taken to be NCE, as it is assumed to be a good proxy of health attentive behavior. == Negative control == Negative controls are variables that meant to help when the study design is suspected to be invalid because of unmeasured confounders that are correlated with both the treatment and the outcome. Where there are only two possible outcomes, e.g. positive or negative, if the treatment group and the negative control (non-treatment group) both produce a negative result, it can be inferred that the treatment had no effect. If the treatment group and the negative control both produce a positive result, it can be inferred that a confounding variable is involved in the phenomenon under study, and the positive results are not solely due to the treatment. In other examples, outcomes might be measured as lengths, times, percentages, and so forth. In the drug testing example, we could measure the percentage of patients cured. In this case, the treatment is inferred to have no effect when the treatment group and the negative control produce the same results. Some improvement is expected in the placebo group due to the placebo effect, and this result sets the baseline upon which the treatment must improve upon. Even if the treatment group shows improvement, it needs to be compared to the placebo group. If the groups show the same effect, then the treatment was not responsible for the improvement (because the same number of patients were cured in the absence of the treatment). The treatment is only effective if the treatment group shows more improvement than the placebo group. === Negative Control Exposure (NCE) === NCE is a variable that should not causally affect the outcome, but may suffer from the same confounding as the exposure-outcome relationship in question. A priori, there should be no statistical association between the NCE and the outcome. If an association is found, then it through the unmeasured confounder, and since the NCE and treatment share the same confounding mechanism, there is an alternative path, apart from the direct path from the treatment to the outcome. In that case, the study design is invalid. For example, Yerushalmy used husband's smoking as an NCE. The exposure was maternal smoking; the outcomes were various birth factors, such as incidence of low birth weight, length of pregnancy, and neonatal mortality rates. It is assumed that husband's smoking share common confounders, such household health lifestyle with the pregnant woman's smoking, but it does not causally affect the fetus development. Nonetheless, Yerushalmy found a statistical association, And as a result, it casts doubt on the proposition that cigarette smoking causally interferes with intrauterine development of the fetus. ==== Differences Between Negative Control Exposures and Placebo ==== The term negative controls is used when the study is based on observations, while the Placebo should be used as a non-treatment in randomized control trials. === Negative Control Outcome (NCO) === Negative Control Outcomes are the more popular type of negative controls. NCO is a variable that is not causally affected by the treatment, but suspected to have a similar confounding mechanism as the treatment-outcome relationship. If the study design is valid, there should be no statistical association between the NCO and the treatment. Thus, an association between them suggest that the design is invalid. For example, Jackson et al. used mortality from all causes outside of influenza season an NCO in a study examining influenza vaccine's effect on influenza-related deaths. A possible confounding mechanism is health status and lifestyle, such as the people who are more healthy in general also tend to take the influenza vaccine. Jackson et al. found that a preferential receipt of vaccine by relatively healthy seniors, and that differences in health status between vaccinated and unvaccinated groups leads to bias in estimates of influenza vaccine effectiveness. In a similar example, when discussing the impact of air pollutants on asthma hospital admissions, Sheppard et al. et al. used non-elderly appendicitis hospital admissions as NCO. ==== Formal Conditions ==== Given a treatment A {\displaystyle A} and an outcome Y {\displaystyle Y} , in the presence of a set of control variables X {\displaystyle X} , and unmeasured confounder U {\displaystyle U} for the A − Y {\displaystyle A-Y} relationship. Shi et al. presented formal conditions for a negative control outcome Y ~ {\displaystyle {\tilde {Y}}} , Stable Unit Treatment Value Assumption (SUTVA): For both Y {\displaystyle {Y}} and Y ~ {\displaystyle {\tilde {Y}}} with regard to A = a {\displaystyle A=a} . Latent Exchangeability: Y A = a ⊥ A | X , U {\displaystyle Y^{A=a}\perp A|\;X,U} Given X {\displaystyle X} and U {\displaystyle U} , the potential outcome Y A = a {\displaystyle Y^{A=a}} is independent of the treatment. Irrelevancy: Ensures the irrelevancy of the treatment on the NCO. Y ~ A = a = Y ~ A = a ′ = Y ~ | U , X {\displaystyle {\tilde {Y}}^{A=a}={\tilde {Y}}^{A=a'}={\tilde {Y}}|\;U,X} : There is no causal effect of A {\displaystyle A} on Y ~ {\displaystyle {\tilde {Y}}} given X {\displaystyle X} and U {\displaystyle U} . Y ~ ⊥ A | U , X {\displaystyle {\tilde {Y}}\perp A|\;U,X} : There is no causal effect of A {\displaystyle A} on Y ~ {\displaystyle {\tilde {Y}}} given X {\displaystyle X} and U {\displaystyle U} . The NCO is independent of the treatment given X {\displaystyle X} and U {\displaystyle U} . U-Comparability: Y ~ ⧸ ⊥ U | X {\displaystyle {\tilde {Y}}\not {\perp }U|\;X} The unmeasured confounders U {\displaystyle U} of the association between A {\displaystyle A} and Y {\displaystyle Y} are the same for the association between A {\displaystyle A} and Y ~ {\displaystyle {\tilde {Y}}} . Given assumption 1 - 4, a non-null association between A {\displaystyle A} and Y ~ {\displaystyle {\tilde {Y}}} , can be explained by U {\displaystyle U} , and not by another mechanism. A possible violation of Latent Exchangeability will be when only the people that are influenced by a medicine will take it, even if both X {\displaystyle X} and U {\displaystyle U} are the same. For example, we would expect that given age and medical history ( X {\displaystyle X} ), general health awareness ( U {\displaystyle U} ), the intake of A {\displaystyle A} influenza vaccine will be independent of potential influenza related deaths Y ~ A = a {\displaystyle {\tilde {Y}}^{A=a}} . Otherwise, the Latent Exchangeability assumption is violated, and no identification can be made. A violation of Irrelevancy occurs when there is a causal effect of A {\displaystyle A} on Y ~ {\displaystyle {\tilde {Y}}} . For example, we would expect that given X {\displaystyle X} and U {\displaystyle U} , the influenza vaccine does not influence all-cause mortality. If, however, during the influenza vaccine medical visit, the physician also performs a general physical test, recommends good health habits, and prescribes vitamins and essential drugs. In this case, there is likely a causal effect of A {\displaystyle A} on Y ~ {\displaystyle {\tilde {Y}}} (conditional on X {\displaystyle X} and U {\displaystyle U} ). Therefore, Y ~ {\displaystyle {\tilde {Y}}} cannot be used as NCO, as the test might fail even if the causal design is valid. U-Comparability is violated when Y ~ ⊥ U {\displaystyle {\tilde {Y}}{\perp }U} , and therefore the lack of association between A {\displaystyle A} and Y ~ {\displaystyle {\tilde {Y}}} does not provide us any evidence for the invalidity of A {\displaystyle A} . This violation would occur when we choose a poor NCO, that is not or very weakly correlated with the unmeasured confounders. == Positive control == Positive controls are often used to assess test validity. For example, to assess a new test's ability to detect a disease (its sensitivity), then we can compare it against a different test that is already known to work. The well-established test is a positive control since we already know that the answer to the question (whether the test works) is yes. Similarly, in an enzyme assay to measure the amount of an enzyme in a set of extracts, a positive control would be an assay containing a known quantity of the purified enzyme (while a negative control would contain no enzyme). The positive control should give a large amount of enzyme activity, while the negative control should give very low to no activity. If the positive control does not produce the expected result, there may be something wrong with the experimental procedure, and the experiment is repeated. For difficult or complicated experiments, the result from the positive control can also help in comparison to previous experimental results. For example, if the well-established disease test was determined to have the same effect as found by previous experimenters, this indicates that the experiment is being performed in the same way that the previous experimenters did. When possible, multiple positive controls may be used—if there is more than one disease test that is known to be effective, more than one might be tested. Multiple positive controls also allow finer comparisons of the results (calibration, or standardization) if the expected results from the positive controls have different sizes. For example, in the enzyme assay discussed above, a standard curve may be produced by making many different samples with different quantities of the enzyme. == Randomization == In randomization, the groups that receive different experimental treatments are determined randomly. While this does not ensure that there are no differences between the groups, it ensures that the differences are distributed equally, thus correcting for systematic errors. For example, in experiments where crop yield is affected (e.g. soil fertility), the experiment can be controlled by assigning the treatments to randomly selected plots of land. This mitigates the effect of variations in soil composition on the yield. == Blind experiments == Blinding is the practice of withholding information that may bias an experiment. For example, participants may not know who received an active treatment and who received a placebo. If this information were to become available to trial participants, patients could receive a larger placebo effect, researchers could influence the experiment to meet their expectations (the observer effect), and evaluators could be subject to confirmation bias. A blind can be imposed on any participant of an experiment, including subjects, researchers, technicians, data analysts, and evaluators. In some cases, sham surgery may be necessary to achieve blinding. During the course of an experiment, a participant becomes unblinded if they deduce or otherwise obtain information that has been masked to them. Unblinding that occurs before the conclusion of a study is a source of experimental error, as the bias that was eliminated by blinding is re-introduced. Unblinding is common in blind experiments and must be measured and reported. Meta-research has revealed high levels of unblinding in pharmacological trials. In particular, antidepressant trials are poorly blinded. Reporting guidelines recommend that all studies assess and report unblinding. In practice, very few studies assess unblinding. Blinding is an important tool of the scientific method, and is used in many fields of research. In some fields, such as medicine, it is considered essential. In clinical research, a trial that is not blinded trial is called an open trial. == See also == False positives and false negatives Designed experiment Controlling for a variable James Lind cured scurvy using a controlled experiment that has been described as the first clinical trial. Randomized controlled trial Wait list control group == References == == External links == "Control" . Encyclopædia Britannica. Vol. 7 (11th ed.). 1911.
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Wikipedia:Negative number#0
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In mathematics, a negative number is the opposite of a positive real number. Equivalently, a negative number is a real number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −(−3) = 3 because the opposite of an opposite is the original value. Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced and read as "minus three" or "negative three". Conversely, a number that is greater than zero is called positive; zero is usually (but not always) thought of as neither positive nor negative. The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign. Every real number other than zero is either positive or negative. The non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3, ...), while the positive and negative whole numbers (together with zero) are referred to as integers. (Some definitions of the natural numbers exclude zero.) In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, as an alternative notation to represent negative numbers. Negative numbers were used in the Nine Chapters on the Mathematical Art, which in its present form dates from the period of the Chinese Han dynasty (202 BC – AD 220), but may well contain much older material. Liu Hui (c. 3rd century) established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers. Islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Prior to the concept of negative numbers, mathematicians such as Diophantus considered negative solutions to problems "false" and equations requiring negative solutions were described as absurd. Western mathematicians like Leibniz held that negative numbers were invalid, but still used them in calculations. == Introduction == === The number line === The relationship between negative numbers, positive numbers, and zero is often expressed in the form of a number line: Numbers appearing farther to the right on this line are greater, while numbers appearing farther to the left are lesser. Thus zero appears in the middle, with the positive numbers to the right and the negative numbers to the left. Note that a negative number with greater magnitude is considered less. For example, even though (positive) 8 is greater than (positive) 5, written negative 8 is considered to be less than negative 5: === Signed numbers === In the context of negative numbers, a number that is greater than zero is referred to as positive. Thus every real number other than zero is either positive or negative, while zero itself is not considered to have a sign. Positive numbers are sometimes written with a plus sign in front, e.g. +3 denotes a positive three. Because zero is neither positive nor negative, the term nonnegative is sometimes used to refer to a number that is either positive or zero, while nonpositive is used to refer to a number that is either negative or zero. Zero is a neutral number. === As the result of subtraction === Negative numbers can be thought of as resulting from the subtraction of a larger number from a smaller. For example, negative three is the result of subtracting three from zero: In general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers. For example, since 8 − 5 = 3. == Everyday uses of negative numbers == === Sport === Goal difference in association football and hockey; points difference in rugby football; net run rate in cricket; golf scores relative to par. Plus-minus differential in ice hockey: the difference in total goals scored for the team (+) and against the team (−) when a particular player is on the ice is the player's +/− rating. Players can have a negative (+/−) rating. Run differential in baseball: the run differential is negative if the team allows more runs than they scored. Clubs may be deducted points for breaches of the laws, and thus have a negative points total until they have earned at least that many points that season. Lap (or sector) times in Formula 1 may be given as the difference compared to a previous lap (or sector) (such as the previous record, or the lap just completed by a driver in front), and will be positive if slower and negative if faster. In some athletics events, such as sprint races, the hurdles, the triple jump and the long jump, the wind assistance is measured and recorded, and is positive for a tailwind and negative for a headwind. === Science === Temperatures which are colder than 0 °C or 0 °F. Latitudes south of the equator and longitudes west of the prime meridian. Topographical features of the earth's surface are given a height above sea level, which can be negative (e.g. the surface elevation of the Dead Sea or Death Valley, or the elevation of the Thames Tideway Tunnel). Electrical circuits. When a battery is connected in reverse polarity, the voltage applied is said to be the opposite of its rated voltage. For example, a 6-volt battery connected in reverse applies a voltage of −6 volts. Ions have a positive or negative electrical charge. Impedance of an AM broadcast tower used in multi-tower directional antenna arrays, which can be positive or negative. === Finance === Financial statements can include negative balances, indicated either by a minus sign or by enclosing the balance in parentheses. Examples include bank account overdrafts and business losses (negative earnings). The annual percentage growth in a country's GDP might be negative, which is one indicator of being in a recession. Occasionally, a rate of inflation may be negative (deflation), indicating a fall in average prices. The daily change in a share price or stock market index, such as the FTSE 100 or the Dow Jones. A negative number in financing is synonymous with "debt" and "deficit" which are also known as "being in the red". Interest rates can be negative, when the lender is charged to deposit their money. === Other === The numbering of stories in a building below the ground floor. When playing an audio file on a portable media player, such as an iPod, the screen display may show the time remaining as a negative number, which increases up to zero time remaining at the same rate as the time already played increases from zero. Television game shows: Participants on QI often finish with a negative points score. Teams on University Challenge have a negative score if their first answers are incorrect and interrupt the question. Jeopardy! has a negative money score – contestants play for an amount of money and any incorrect answer that costs them more than what they have now can result in a negative score. In The Price Is Right's pricing game Buy or Sell, if an amount of money is lost that is more than the amount currently in the bank, it incurs a negative score. The change in support for a political party between elections, known as swing. A politician's approval rating. In video games, a negative number indicates loss of life, damage, a score penalty, or consumption of a resource, depending on the genre of the simulation. Employees with flexible working hours may have a negative balance on their timesheet if they have worked fewer total hours than contracted to that point. Employees may be able to take more than their annual holiday allowance in a year, and carry forward a negative balance to the next year. Transposing notes on an electronic keyboard are shown on the display with positive numbers for increases and negative numbers for decreases, e.g. "−1" for one semitone down. == Arithmetic involving negative numbers == The minus sign "−" signifies the operator for both the binary (two-operand) operation of subtraction (as in y − z) and the unary (one-operand) operation of negation (as in −x, or twice in −(−x)). A special case of unary negation occurs when it operates on a positive number, in which case the result is a negative number (as in −5). The ambiguity of the "−" symbol does not generally lead to ambiguity in arithmetical expressions, because the order of operations makes only one interpretation or the other possible for each "−". However, it can lead to confusion and be difficult for a person to understand an expression when operator symbols appear adjacent to one another. A solution can be to parenthesize the unary "−" along with its operand. For example, the expression 7 + −5 may be clearer if written 7 + (−5) (even though they mean exactly the same thing formally). The subtraction expression 7 − 5 is a different expression that doesn't represent the same operations, but it evaluates to the same result. Sometimes in elementary schools a number may be prefixed by a superscript minus sign or plus sign to explicitly distinguish negative and positive numbers as in === Addition === Addition of two negative numbers is very similar to addition of two positive numbers. For example, The idea is that two debts can be combined into a single debt of greater magnitude. When adding together a mixture of positive and negative numbers, one can think of the negative numbers as positive quantities being subtracted. For example: In the first example, a credit of 8 is combined with a debt of 3, which yields a total credit of 5. If the negative number has greater magnitude, then the result is negative: Here the credit is less than the debt, so the net result is a debt. === Subtraction === As discussed above, it is possible for the subtraction of two non-negative numbers to yield a negative answer: In general, subtraction of a positive number yields the same result as the addition of a negative number of equal magnitude. Thus and On the other hand, subtracting a negative number yields the same result as the addition a positive number of equal magnitude. (The idea is that losing a debt is the same thing as gaining a credit.) Thus and === Multiplication === When multiplying numbers, the magnitude of the product is always just the product of the two magnitudes. The sign of the product is determined by the following rules: The product of one positive number and one negative number is negative. The product of two negative numbers is positive. Thus and The reason behind the first example is simple: adding three −2's together yields −6: The reasoning behind the second example is more complicated. The idea again is that losing a debt is the same thing as gaining a credit. In this case, losing two debts of three each is the same as gaining a credit of six: The convention that a product of two negative numbers is positive is also necessary for multiplication to follow the distributive law. In this case, we know that Since 2 × (−3) = −6, the product (−2) × (−3) must equal 6. These rules lead to another (equivalent) rule—the sign of any product a × b depends on the sign of a as follows: if a is positive, then the sign of a × b is the same as the sign of b, and if a is negative, then the sign of a × b is the opposite of the sign of b. The justification for why the product of two negative numbers is a positive number can be observed in the analysis of complex numbers. === Division === The sign rules for division are the same as for multiplication. For example, and If dividend and divisor have the same sign, the result is positive, if they have different signs the result is negative. == Negation == The negative version of a positive number is referred to as its negation. For example, −3 is the negation of the positive number 3. The sum of a number and its negation is equal to zero: That is, the negation of a positive number is the additive inverse of the number. Using algebra, we may write this principle as an algebraic identity: This identity holds for any positive number x. It can be made to hold for all real numbers by extending the definition of negation to include zero and negative numbers. Specifically: The negation of 0 is 0, and The negation of a negative number is the corresponding positive number. For example, the negation of −3 is +3. In general, The absolute value of a number is the non-negative number with the same magnitude. For example, the absolute value of −3 and the absolute value of 3 are both equal to 3, and the absolute value of 0 is 0. == Formal construction of negative integers == In a similar manner to rational numbers, we can extend the natural numbers N {\displaystyle \mathbb {N} } to the integers Z {\displaystyle \mathbb {Z} } by defining integers as an ordered pair of natural numbers (a, b). We can extend addition and multiplication to these pairs with the following rules: We define an equivalence relation ~ upon these pairs with the following rule: This equivalence relation is compatible with the addition and multiplication defined above, and we may define Z {\displaystyle \mathbb {Z} } to be the quotient set N 2 / ∼ {\displaystyle \mathbb {N} ^{2}/\sim } , i.e. we identify two pairs (a, b) and (c, d) if they are equivalent in the above sense. Note that Z {\displaystyle \mathbb {Z} } , equipped with these operations of addition and multiplication, is a ring, and is in fact, the prototypical example of a ring. We can also define a total order on Z {\displaystyle \mathbb {Z} } by writing This will lead to an additive zero of the form (a, a), an additive inverse of (a, b) of the form (b, a), a multiplicative unit of the form (a + 1, a), and a definition of subtraction This construction is a special case of the Grothendieck construction. === Uniqueness === The additive inverse of a number is unique, as is shown by the following proof. As mentioned above, an additive inverse of a number is defined as a value which when added to the number yields zero. Let x be a number and let y be its additive inverse. Suppose y′ is another additive inverse of x. By definition, x + y ′ = 0 , and x + y = 0. {\displaystyle x+y'=0,\quad {\text{and}}\quad x+y=0.} And so, x + y′ = x + y. Using the law of cancellation for addition, it is seen that y′ = y. Thus y is equal to any other additive inverse of x. That is, y is the unique additive inverse of x. == History == For a long time, understanding of negative numbers was delayed by the impossibility of having a negative-number amount of a physical object, for example "minus-three apples", and negative solutions to problems were considered "false". In Hellenistic Egypt, the Greek mathematician Diophantus in the 3rd century AD referred to an equation that was equivalent to 4 x + 20 = 4 {\displaystyle 4x+20=4} (which has a negative solution) in Arithmetica, saying that the equation was absurd. For this reason Greek geometers were able to solve geometrically all forms of the quadratic equation which give positive roots, while they could take no account of others. Negative numbers appear for the first time in history in the Nine Chapters on the Mathematical Art (九章算術, Jiǔ zhāng suàn-shù), which in its present form dates from the Han period, but may well contain much older material. The mathematician Liu Hui (c. 3rd century) established rules for the addition and subtraction of negative numbers. The historian Jean-Claude Martzloff theorized that the importance of duality in Chinese natural philosophy made it easier for the Chinese to accept the idea of negative numbers. The Chinese were able to solve simultaneous equations involving negative numbers. The Nine Chapters used red counting rods to denote positive coefficients and black rods for negative. This system is the exact opposite of contemporary printing of positive and negative numbers in the fields of banking, accounting, and commerce, wherein red numbers denote negative values and black numbers signify positive values. Liu Hui writes: Now there are two opposite kinds of counting rods for gains and losses, let them be called positive and negative. Red counting rods are positive, black counting rods are negative. The ancient Indian Bakhshali Manuscript carried out calculations with negative numbers, using "+" as a negative sign. The date of the manuscript is uncertain. LV Gurjar dates it no later than the 4th century, Hoernle dates it between the third and fourth centuries, Ayyangar and Pingree dates it to the 8th or 9th centuries, and George Gheverghese Joseph dates it to about AD 400 and no later than the early 7th century. During the 7th century AD, negative numbers were used in India to represent debts. The Indian mathematician Brahmagupta, in Brahma-Sphuta-Siddhanta (written c. AD 630), discussed the use of negative numbers to produce a general form quadratic formula similar to the one in use today. In the 9th century, Islamic mathematicians were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid. Al-Khwarizmi in his Al-jabr wa'l-muqabala (from which the word "algebra" derives) did not use negative numbers or negative coefficients. But within fifty years, Abu Kamil illustrated the rules of signs for expanding the multiplication ( a ± b ) ( c ± d ) {\displaystyle (a\pm b)(c\pm d)} , and al-Karaji wrote in his al-Fakhrī that "negative quantities must be counted as terms". In the 10th century, Abū al-Wafā' al-Būzjānī considered debts as negative numbers in A Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen. By the 12th century, al-Karaji's successors were to state the general rules of signs and use them to solve polynomial divisions. As al-Samaw'al writes: the product of a negative number—al-nāqiṣ (loss)—by a positive number—al-zāʾid (gain)—is negative, and by a negative number is positive. If we subtract a negative number from a higher negative number, the remainder is their negative difference. The difference remains positive if we subtract a negative number from a lower negative number. If we subtract a negative number from a positive number, the remainder is their positive sum. If we subtract a positive number from an empty power (martaba khāliyya), the remainder is the same negative, and if we subtract a negative number from an empty power, the remainder is the same positive number. In the 12th century in India, Bhāskara II gave negative roots for quadratic equations but rejected them because they were inappropriate in the context of the problem. He stated that a negative value is "in this case not to be taken, for it is inadequate; people do not approve of negative roots." Fibonacci allowed negative solutions in financial problems where they could be interpreted as debits (chapter 13 of Liber Abaci, 1202) and later as losses (in Flos, 1225). In the 15th century, Nicolas Chuquet, a Frenchman, used negative numbers as exponents but referred to them as "absurd numbers". Michael Stifel dealt with negative numbers in his 1544 AD Arithmetica Integra, where he also called them numeri absurdi (absurd numbers). In 1545, Gerolamo Cardano, in his Ars Magna, provided the first satisfactory treatment of negative numbers in Europe. He did not allow negative numbers in his consideration of cubic equations, so he had to treat, for example, x 3 + a x = b {\displaystyle x^{3}+ax=b} separately from x 3 = a x + b {\displaystyle x^{3}=ax+b} (with a , b > 0 {\displaystyle a,b>0} in both cases). In all, Cardano was driven to the study of thirteen types of cubic equations, each with all negative terms moved to the other side of the = sign to make them positive. (Cardano also dealt with complex numbers, but understandably liked them even less.) == See also == == References == === Citations === === Bibliography === == External links == Maseres' biographical information BBC Radio 4 series In Our Time, on "Negative Numbers", 9 March 2006 Endless Examples & Exercises: Operations with Signed Integers Math Forum: Ask Dr. Math FAQ: Negative Times a Negative
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Wikipedia:Negligible function#0
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In mathematics, a negligible function is a function μ : N → R {\displaystyle \mu :\mathbb {N} \to \mathbb {R} } such that for every positive integer c there exists an integer Nc such that for all x > Nc, | μ ( x ) | < 1 x c . {\displaystyle |\mu (x)|<{\frac {1}{x^{c}}}.} Equivalently, we may also use the following definition. A function μ : N → R {\displaystyle \mu :\mathbb {N} \to \mathbb {R} } is negligible, if for every positive polynomial poly(·) there exists an integer Npoly > 0 such that for all x > Npoly | μ ( x ) | < 1 poly ( x ) . {\displaystyle |\mu (x)|<{\frac {1}{\operatorname {poly} (x)}}.} == History == The concept of negligibility can find its trace back to sound models of analysis. Though the concepts of "continuity" and "infinitesimal" became important in mathematics during Newton and Leibniz's time (1680s), they were not well-defined until the late 1810s. The first reasonably rigorous definition of continuity in mathematical analysis was due to Bernard Bolzano, who wrote in 1817 the modern definition of continuity. Later Cauchy, Weierstrass and Heine also defined as follows (with all numbers in the real number domain R {\displaystyle \mathbb {R} } ): (Continuous function) A function f : R → R {\displaystyle f:\mathbb {R} {\rightarrow }\mathbb {R} } is continuous at x = x 0 {\displaystyle x=x_{0}} if for every ε > 0 {\displaystyle \varepsilon >0} , there exists a positive number δ > 0 {\displaystyle \delta >0} such that | x − x 0 | < δ {\displaystyle |x-x_{0}|<\delta } implies | f ( x ) − f ( x 0 ) | < ε . {\displaystyle |f(x)-f(x_{0})|<\varepsilon .} This classic definition of continuity can be transformed into the definition of negligibility in a few steps by changing parameters used in the definition. First, in the case x 0 = ∞ {\displaystyle x_{0}=\infty } with f ( x 0 ) = 0 {\displaystyle f(x_{0})=0} , we must define the concept of "infinitesimal function": (Infinitesimal) A continuous function μ : R → R {\displaystyle \mu :\mathbb {R} \to \mathbb {R} } is infinitesimal (as x {\displaystyle x} goes to infinity) if for every ε > 0 {\displaystyle \varepsilon >0} there exists N ε {\displaystyle N_{\varepsilon }} such that for all x > N ε {\displaystyle x>N_{\varepsilon }} | μ ( x ) | < ε . {\displaystyle |\mu (x)|<\varepsilon \,.} Next, we replace ε > 0 {\displaystyle \varepsilon >0} by the functions 1 / x c {\displaystyle 1/x^{c}} where c > 0 {\displaystyle c>0} or by 1 / poly ( x ) {\displaystyle 1/\operatorname {poly} (x)} where poly ( x ) {\displaystyle \operatorname {poly} (x)} is a positive polynomial. This leads to the definitions of negligible functions given at the top of this article. Since the constants ε > 0 {\displaystyle \varepsilon >0} can be expressed as 1 / poly ( x ) {\displaystyle 1/\operatorname {poly} (x)} with a constant polynomial, this shows that infinitesimal functions are a superset of negligible functions. == Use in cryptography == In complexity-based modern cryptography, a security scheme is provably secure if the probability of security failure (e.g., inverting a one-way function, distinguishing cryptographically strong pseudorandom bits from truly random bits) is negligible in terms of the input x {\displaystyle x} = cryptographic key length n {\displaystyle n} . Hence comes the definition at the top of the page because key length n {\displaystyle n} must be a natural number. Nevertheless, the general notion of negligibility doesn't require that the input parameter x {\displaystyle x} is the key length n {\displaystyle n} . Indeed, x {\displaystyle x} can be any predetermined system metric and corresponding mathematical analysis would illustrate some hidden analytical behaviors of the system. The reciprocal-of-polynomial formulation is used for the same reason that computational boundedness is defined as polynomial running time: it has mathematical closure properties that make it tractable in the asymptotic setting (see #Closure properties). For example, if an attack succeeds in violating a security condition only with negligible probability, and the attack is repeated a polynomial number of times, the success probability of the overall attack still remains negligible. In practice one might want to have more concrete functions bounding the adversary's success probability and to choose the security parameter large enough that this probability is smaller than some threshold, say 2−128. == Closure properties == One of the reasons that negligible functions are used in foundations of complexity-theoretic cryptography is that they obey closure properties. Specifically, If f , g : N → R {\displaystyle f,g:\mathbb {N} \to \mathbb {R} } are negligible, then the function x ↦ f ( x ) + g ( x ) {\displaystyle x\mapsto f(x)+g(x)} is negligible. If f : N → R {\displaystyle f:\mathbb {N} \to \mathbb {R} } is negligible and p {\displaystyle p} is any real polynomial, then the function x ↦ p ( x ) ⋅ f ( x ) {\displaystyle x\mapsto p(x)\cdot f(x)} is negligible. Conversely, if f : N → R {\displaystyle f:\mathbb {N} \to \mathbb {R} } is not negligible, then neither is x ↦ f ( x ) / p ( x ) {\displaystyle x\mapsto f(x)/p(x)} for any real polynomial p {\displaystyle p} . == Examples == n ↦ a − n {\displaystyle n\mapsto a^{-n}} is negligible for any a ≥ 2 {\displaystyle a\geq 2} : Step: This is an exponential decay function where a {\displaystyle a} is a constant greater than or equal to 2. As n → ∞ {\displaystyle n\to \infty } , a − n → 0 {\displaystyle a^{-n}\to 0} very quickly, making it negligible. f ( n ) = 3 − n {\displaystyle f(n)=3^{-{\sqrt {n}}}} is negligible: Step: This function has exponential decay with a base of 3, but the exponent grows slower than n {\displaystyle n} (only at n {\displaystyle {\sqrt {n}}} ). As n → ∞ {\displaystyle n\to \infty } , 3 − n → 0 {\displaystyle 3^{-{\sqrt {n}}}\to 0} , so it’s still negligible but decays slower than 3 − n {\displaystyle 3^{-n}} . f ( n ) = n − log n {\displaystyle f(n)=n^{-\log n}} is negligible: Step: In this case, n − log n {\displaystyle n^{-\log n}} represents a polynomial decay, with the exponent growing negatively due to log n {\displaystyle \log n} . Since the decay rate increases with n {\displaystyle n} , the function goes to 0 faster than polynomial functions like n − k {\displaystyle n^{-k}} for any constant k {\displaystyle k} , making it negligible. f ( n ) = ( log n ) − log n {\displaystyle f(n)=(\log n)^{-\log n}} is negligible: Step: This function decays as the logarithm of n {\displaystyle n} raised to a negative exponent − log n {\displaystyle -\log n} , which leads to a fast approach to 0 as n → ∞ {\displaystyle n\to \infty } . The decay here is faster than inverse logarithmic or polynomial rates, making it negligible. f ( n ) = 2 − c log n {\displaystyle f(n)=2^{-c\log n}} is not negligible, for positive c {\displaystyle c} : Step: We can rewrite this as f ( n ) = n − c {\displaystyle f(n)=n^{-c}} , which is a polynomial decay rather than an exponential one. Since c {\displaystyle c} is positive, f ( n ) → 0 {\displaystyle f(n)\to 0} as n → ∞ {\displaystyle n\to \infty } , but it doesn’t decay as quickly as true exponential functions with respect to n {\displaystyle n} , making it non-negligible. Assume n > 0 {\displaystyle n>0} , we take the limit as n → ∞ {\displaystyle n\to \infty } : Negligible: f ( n ) = 1 x n / 2 {\displaystyle f(n)={\frac {1}{x^{n/2}}}} : Step: This function decays exponentially with base x {\displaystyle x} raised to the power of − n 2 {\displaystyle -{\frac {n}{2}}} . As n → ∞ {\displaystyle n\to \infty } , x − n 2 → 0 {\displaystyle x^{-{\frac {n}{2}}}\to 0} quickly, making it negligible. f ( n ) = 1 x log ( n k ) {\displaystyle f(n)={\frac {1}{x^{\log {(n^{k})}}}}} for k ≥ 1 {\displaystyle k\geq 1} : Step: We can simplify x − log ( n k ) {\displaystyle x^{-\log(n^{k})}} as n − k log x {\displaystyle n^{-k\log x}} , which decays faster than any polynomial. As n → ∞ {\displaystyle n\to \infty } , the function approaches zero and is considered negligible for any k ≥ 1 {\displaystyle k\geq 1} and x > 1 {\displaystyle x>1} . f ( n ) = 1 x ( log n ) k {\displaystyle f(n)={\frac {1}{x^{(\log n)^{k}}}}} for k ≥ 1 {\displaystyle k\geq 1} : Step: The decay is determined by the base x {\displaystyle x} raised to the power of − ( log n ) k {\displaystyle -(\log n)^{k}} . Since ( log n ) k {\displaystyle (\log n)^{k}} grows with n {\displaystyle n} , this function approaches zero faster than polynomial decay, making it negligible.= f ( n ) = 1 x n {\displaystyle f(n)={\frac {1}{x^{\sqrt {n}}}}} : Step: Here, f ( n ) {\displaystyle f(n)} decays exponentially with a base of x {\displaystyle x} raised to − n {\displaystyle -{\sqrt {n}}} . As n → ∞ {\displaystyle n\to \infty } , f ( n ) → 0 {\displaystyle f(n)\to 0} quickly, so it’s considered negligible. Non-negligible: f ( n ) = 1 n 1 / n {\displaystyle f(n)={\frac {1}{n^{1/n}}}} : Step: Since n 1 / n → 1 {\displaystyle n^{1/n}\to 1} as n → ∞ {\displaystyle n\to \infty } , this function decays very slowly, failing to approach zero quickly enough to be considered negligible. f ( n ) = 1 x n ( log n ) {\displaystyle f(n)={\frac {1}{x^{n(\log n)}}}} : Step: With an exponential base and exponent n ( log n ) {\displaystyle n(\log n)} , this function would approach zero very rapidly, suggesting negligibility. == See also == Negligible set Colombeau algebra Nonstandard numbers Gromov's theorem on groups of polynomial growth Non-standard calculus == References == Goldreich, Oded (2001). Foundations of Cryptography: Volume 1, Basic Tools. Cambridge University Press. ISBN 0-521-79172-3. Sipser, Michael (1997). "Section 10.6.3: One-way functions". Introduction to the Theory of Computation. PWS Publishing. pp. 374–376. ISBN 0-534-94728-X. Papadimitriou, Christos (1993). "Section 12.1: One-way functions". Computational Complexity (1st ed.). Addison Wesley. pp. 279–298. ISBN 0-201-53082-1. Colombeau, Jean François (1984). New Generalized Functions and Multiplication of Distributions. Mathematics Studies 84, North Holland. ISBN 0-444-86830-5. Bellare, Mihir (1997). "A Note on Negligible Functions". Journal of Cryptology. 15. Dept. of Computer Science & Engineering University of California at San Diego: 2002. CiteSeerX 10.1.1.43.7900.
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Wikipedia:Negligible set#0
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In mathematics, a negligible set is a set that is small enough that it can be ignored for some purpose. As common examples, finite sets can be ignored when studying the limit of a sequence, and null sets can be ignored when studying the integral of a measurable function. Negligible sets define several useful concepts that can be applied in various situations, such as truth almost everywhere. In order for these to work, it is generally only necessary that the negligible sets form an ideal; that is, that the empty set be negligible, the union of two negligible sets be negligible, and any subset of a negligible set be negligible. For some purposes, we also need this ideal to be a sigma-ideal, so that countable unions of negligible sets are also negligible. If I and J are both ideals of subsets of the same set X, then one may speak of I-negligible and J-negligible subsets. The opposite of a negligible set is a generic property, which has various forms. == Examples == Let X be the set N of natural numbers, and let a subset of N be negligible if it is finite. Then the negligible sets form an ideal. This idea can be applied to any infinite set; but if applied to a finite set, every subset will be negligible, which is not a very useful notion. Or let X be an uncountable set, and let a subset of X be negligible if it is countable. Then the negligible sets form a sigma-ideal. Let X be a measurable space equipped with a measure m, and let a subset of X be negligible if it is m-null. Then the negligible sets form a sigma-ideal. Every sigma-ideal on X can be recovered in this way by placing a suitable measure on X, although the measure may be rather pathological. Let X be the set R of real numbers, and let a subset A of R be negligible if for each ε > 0, there exists a finite or countable collection I1, I2, … of (possibly overlapping) intervals satisfying: A ⊂ ⋃ k I k {\displaystyle A\subset \bigcup _{k}I_{k}} and ∑ k | I k | < ϵ . {\displaystyle \sum _{k}|I_{k}|<\epsilon .} This is a special case of the preceding example, using Lebesgue measure, but described in elementary terms. Let X be a topological space, and let a subset be negligible if it is of first category, that is, if it is a countable union of nowhere-dense sets (where a set is nowhere-dense if it is not dense in any open set). Then the negligible sets form a sigma-ideal. Let X be a directed set, and let a subset of X be negligible if it has an upper bound. Then the negligible sets form an ideal. The first example is a special case of this using the usual ordering of N. In a coarse structure, the controlled sets are negligible. == Derived concepts == Let X be a set, and let I be an ideal of negligible subsets of X. If p is a proposition about the elements of X, then p is true almost everywhere if the set of points where p is true is the complement of a negligible set. That is, p may not always be true, but it's false so rarely that this can be ignored for the purposes at hand. If f and g are functions from X to the same space Y, then f and g are equivalent if they are equal almost everywhere. To make the introductory paragraph precise, then, let X be N, and let the negligible sets be the finite sets. Then f and g are sequences. If Y is a topological space, then f and g have the same limit, or both have none. (When you generalise this to a directed sets, you get the same result, but for nets.) Or, let X be a measure space, and let negligible sets be the null sets. If Y is the real line R, then either f and g have the same integral, or neither integral is defined. == See also == Negligible function Generic property == References ==
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Wikipedia:Neighbourhood (mathematics)#0
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In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space. It is closely related to the concepts of open set and interior. Intuitively speaking, a neighbourhood of a point is a set of points containing that point where one can move some amount in any direction away from that point without leaving the set. == Definitions == === Neighbourhood of a point === If X {\displaystyle X} is a topological space and p {\displaystyle p} is a point in X , {\displaystyle X,} then a neighbourhood of p {\displaystyle p} is a subset V {\displaystyle V} of X {\displaystyle X} that includes an open set U {\displaystyle U} containing p {\displaystyle p} , p ∈ U ⊆ V ⊆ X . {\displaystyle p\in U\subseteq V\subseteq X.} This is equivalent to the point p ∈ X {\displaystyle p\in X} belonging to the topological interior of V {\displaystyle V} in X . {\displaystyle X.} The neighbourhood V {\displaystyle V} need not be an open subset of X . {\displaystyle X.} When V {\displaystyle V} is open (resp. closed, compact, etc.) in X , {\displaystyle X,} it is called an open neighbourhood (resp. closed neighbourhood, compact neighbourhood, etc.). Some authors require neighbourhoods to be open, so it is important to note their conventions. A set that is a neighbourhood of each of its points is open since it can be expressed as the union of open sets containing each of its points. A closed rectangle, as illustrated in the figure, is not a neighbourhood of all its points; points on the edges or corners of the rectangle are not contained in any open set that is contained within the rectangle. The collection of all neighbourhoods of a point is called the neighbourhood system at the point. === Neighbourhood of a set === If S {\displaystyle S} is a subset of a topological space X {\displaystyle X} , then a neighbourhood of S {\displaystyle S} is a set V {\displaystyle V} that includes an open set U {\displaystyle U} containing S {\displaystyle S} , S ⊆ U ⊆ V ⊆ X . {\displaystyle S\subseteq U\subseteq V\subseteq X.} It follows that a set V {\displaystyle V} is a neighbourhood of S {\displaystyle S} if and only if it is a neighbourhood of all the points in S . {\displaystyle S.} Furthermore, V {\displaystyle V} is a neighbourhood of S {\displaystyle S} if and only if S {\displaystyle S} is a subset of the interior of V . {\displaystyle V.} A neighbourhood of S {\displaystyle S} that is also an open subset of X {\displaystyle X} is called an open neighbourhood of S . {\displaystyle S.} The neighbourhood of a point is just a special case of this definition. == In a metric space == In a metric space M = ( X , d ) , {\displaystyle M=(X,d),} a set V {\displaystyle V} is a neighbourhood of a point p {\displaystyle p} if there exists an open ball with center p {\displaystyle p} and radius r > 0 , {\displaystyle r>0,} such that B r ( p ) = B ( p ; r ) = { x ∈ X : d ( x , p ) < r } {\displaystyle B_{r}(p)=B(p;r)=\{x\in X:d(x,p)<r\}} is contained in V . {\displaystyle V.} V {\displaystyle V} is called a uniform neighbourhood of a set S {\displaystyle S} if there exists a positive number r {\displaystyle r} such that for all elements p {\displaystyle p} of S , {\displaystyle S,} B r ( p ) = { x ∈ X : d ( x , p ) < r } {\displaystyle B_{r}(p)=\{x\in X:d(x,p)<r\}} is contained in V . {\displaystyle V.} Under the same condition, for r > 0 , {\displaystyle r>0,} the r {\displaystyle r} -neighbourhood S r {\displaystyle S_{r}} of a set S {\displaystyle S} is the set of all points in X {\displaystyle X} that are at distance less than r {\displaystyle r} from S {\displaystyle S} (or equivalently, S r {\displaystyle S_{r}} is the union of all the open balls of radius r {\displaystyle r} that are centered at a point in S {\displaystyle S} ): S r = ⋃ p ∈ S B r ( p ) . {\displaystyle S_{r}=\bigcup \limits _{p\in {}S}B_{r}(p).} It directly follows that an r {\displaystyle r} -neighbourhood is a uniform neighbourhood, and that a set is a uniform neighbourhood if and only if it contains an r {\displaystyle r} -neighbourhood for some value of r . {\displaystyle r.} == Examples == Given the set of real numbers R {\displaystyle \mathbb {R} } with the usual Euclidean metric and a subset V {\displaystyle V} defined as V := ⋃ n ∈ N B ( n ; 1 / n ) , {\displaystyle V:=\bigcup _{n\in \mathbb {N} }B\left(n\,;\,1/n\right),} then V {\displaystyle V} is a neighbourhood for the set N {\displaystyle \mathbb {N} } of natural numbers, but is not a uniform neighbourhood of this set. == Topology from neighbourhoods == The above definition is useful if the notion of open set is already defined. There is an alternative way to define a topology, by first defining the neighbourhood system, and then open sets as those sets containing a neighbourhood of each of their points. A neighbourhood system on X {\displaystyle X} is the assignment of a filter N ( x ) {\displaystyle N(x)} of subsets of X {\displaystyle X} to each x {\displaystyle x} in X , {\displaystyle X,} such that the point x {\displaystyle x} is an element of each U {\displaystyle U} in N ( x ) {\displaystyle N(x)} each U {\displaystyle U} in N ( x ) {\displaystyle N(x)} contains some V {\displaystyle V} in N ( x ) {\displaystyle N(x)} such that for each y {\displaystyle y} in V , {\displaystyle V,} U {\displaystyle U} is in N ( y ) . {\displaystyle N(y).} One can show that both definitions are compatible, that is, the topology obtained from the neighbourhood system defined using open sets is the original one, and vice versa when starting out from a neighbourhood system. == Uniform neighbourhoods == In a uniform space S = ( X , Φ ) , {\displaystyle S=(X,\Phi ),} V {\displaystyle V} is called a uniform neighbourhood of P {\displaystyle P} if there exists an entourage U ∈ Φ {\displaystyle U\in \Phi } such that V {\displaystyle V} contains all points of X {\displaystyle X} that are U {\displaystyle U} -close to some point of P ; {\displaystyle P;} that is, U [ x ] ⊆ V {\displaystyle U[x]\subseteq V} for all x ∈ P . {\displaystyle x\in P.} == Deleted neighbourhood == A deleted neighbourhood of a point p {\displaystyle p} (sometimes called a punctured neighbourhood) is a neighbourhood of p , {\displaystyle p,} without { p } . {\displaystyle \{p\}.} For instance, the interval ( − 1 , 1 ) = { y : − 1 < y < 1 } {\displaystyle (-1,1)=\{y:-1<y<1\}} is a neighbourhood of p = 0 {\displaystyle p=0} in the real line, so the set ( − 1 , 0 ) ∪ ( 0 , 1 ) = ( − 1 , 1 ) ∖ { 0 } {\displaystyle (-1,0)\cup (0,1)=(-1,1)\setminus \{0\}} is a deleted neighbourhood of 0. {\displaystyle 0.} A deleted neighbourhood of a given point is not in fact a neighbourhood of the point. The concept of deleted neighbourhood occurs in the definition of the limit of a function and in the definition of limit points (among other things). == See also == Isolated point – Point of a subset S around which there are no other points of S Neighbourhood system – Concept in mathematics Region (mathematics) – Connected open subset of a topological spacePages displaying short descriptions of redirect targets Tubular neighbourhood – neighborhood of a submanifold homeomorphic to that submanifold’s normal bundlePages displaying wikidata descriptions as a fallback == Notes == == References == Bredon, Glen E. (1993). Topology and geometry. New York: Springer-Verlag. ISBN 0-387-97926-3. Engelking, Ryszard (1989). General Topology. Heldermann Verlag, Berlin. ISBN 3-88538-006-4. Kaplansky, Irving (2001). Set Theory and Metric Spaces. American Mathematical Society. ISBN 0-8218-2694-8. Kelley, John L. (1975). General topology. New York: Springer-Verlag. ISBN 0-387-90125-6. Willard, Stephen (2004) [1970]. General Topology. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.
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Wikipedia:Nelly Litvak#0
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Nelly Vladimirovna Litvak (Russian: Нелли Владимировна Литвак, born January 27, 1972) is a Russian and Dutch applied mathematician whose research includes the study of complex networks, stochastic processes, and their applications in medical logistics. Formerly a professor at the University of Twente, she moved to the Eindhoven University of Technology in 2023. == Education and career == Nelly Litvak was born in Gorky (now Nizhny Novgorod), Russia to the journalist Nina Zvereva and the researcher Vladimir Antonets. In 1995 Litvak graduated from the N. I. Lobachevsky State University of Nizhny Novgorod, and in 1998 she earned the candidate's degree in physical and mathematical sciences from the latter university with the dissertation Adaptive Control of Conflicting Flows supervised by Mikhail Andreevich Fedotkin. In 1999 she moved to the Netherlands. She completed another doctoral degree in 2002, a Ph.D. from the Eindhoven University of Technology, with the dissertation Collecting n {\displaystyle n} Items Randomly Located on a Circle, completed in EURANDOM, jointly promoted by Ivo Adan, Jaap Wessels, and Henk Zijm. She became a lecturer at the University of Twente in 2002, and was promoted to associate professor in 2011 and full professor in 2018. In 2017, she took on a second part-time affiliation with the Eindhoven University of Technology, and in 2023 she moved from Twente to a full-time position at the Eindhoven University of Technology, as professor of algorithms for complex networks. == Books == Litvak is the author of several popular science books, including: Наши хорошие подростки (Our good teenagers, in Russian, Alpina, 2010) Формула призвания – 7 правил выбора вуза (Vocation formula – 7 rules for choosing university, Alpina, 2012) IQ to Love: What makes highly intelligent men attractive to women (self-published, 2014) Кому нужна математика? Понятная книга о том, как устроен цифровой мир (Who Needs Mathematics? A Clear Book about how the Digital World Works, with Andrey Raygorodsky, in Russian, Mann-Ivanov-Ferber («Манн, Иванов и Фербер»), 2017) The book was shortlisted for the Russian 2017 Enlightener Prize. Математика для безнадежных гуманитариев (Mathematics for hopeless humanities geeks, with Alla Kechedzhan, in Russian, AST, 2019) == Recognition == Litvak was awarded the 2002 Stieltjes Prize for her Ph.D. dissertation. She was the 2011 recipient of the Professor De Winter award of the University of Twente, given annually to recognize the research of a female faculty member. She was named the university's teacher of the year in 2022. == Personal == While in the Netherlands, Nelly Litvak married to Pranab Mandal. She has two daughters: Natalia (born in 1993 from the first marriage) and Piyali (born in 2005 to Nelly and Pranab). She has younger sister Yekaterina and brother Pyotr. == References == == External links == Nelly Litvak publications indexed by Google Scholar Old profile @ UTwente (unmaintained since 2019) Current profile @ UTwente Profile @ TU/E
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Wikipedia:Nels David Nelson#0
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(Nels) David Nelson, an American mathematician and logician, was born on January 2, 1918, in Cape Girardeau, Missouri. Upon graduation from the Ph.D. program at the University of Wisconsin-Madison, Nelson relocated to Washington, D.C. Nelson remained in Washington, D.C. as a Professor of Mathematics at The George Washington University until his death on August 22, 2003. == Education == David Nelson completed his undergraduate and graduate coursework at the University of Wisconsin-Madison in 1939 and 1940, respectively.[1] Nelson completed his Ph.D. at Madison in 1946. His dissertation, entitled "Recursive Functions and Intuitionistic Number Theory," served as the capstone project for his doctorate. Fellow mathematician Stephen Cole Kleene served as Nelson's doctoral advisor. Nelson, consequently, was Kleene's first doctoral student.[2] According to the Association for Symbolic Logic: Nelson's research was in the area of intuitionistic logic and its connection with recursive function theory. He investigated the relationship, in intuitionistic formal systems, between a truth definition and the provability of formulas representing statements of number theory. Kleene had previously introduced the intuitionistic truth definition and arithmetized this truth notion in his definition of realizability of a formula by a number. As a consequence, they demonstrated that certain classically true formulas are unverifiable in the intuitionistic predicate calculus with strong negation.[3] == Professional career == Nelson taught at Amherst College from 1942 to 1946 as an assistant professor. Upon completion of his doctoral studies, Nelson accepted an assistant professor position with the Department of Mathematics at The George Washington University in Washington, D.C., in 1946. Nelson was officially promoted to the position of professor in 1958. After a decade of service to the university, Nelson received chairmanship of the Department of Mathematics, a position which he held from 1956 to 1967.[4] == Publications == Nelson, David (16 May 1949). "Constructible Falsity". Journal of Symbolic Logic. 14 (1): 16–26. doi:10.2307/2268973. JSTOR 2268973. This paper dealt with the issues of constructive logic in relation to intuitionistic truth. — (December 1966). "Non-null Implication". Journal of Symbolic Logic. 31 (4): 562–572. doi:10.2307/2269691. JSTOR 2269691. == Students == David Nelson oversaw the dissertation work of the George Washington University student John Kent Minichiello, who authored "Negationless Intuitionistic Mathematics" in 1967. Minichiello received the Ruggles Prize for Mathematics in 1963 for excellence in mathematics under the direction of Nelson. == Associations and memberships == Member, Executive Committee of the Association for Symbolic Logic, 1949–1953. [5] Consultant, National Research Council, 1960–1963. [6] == Notes == == References == Minichiello, J. Kent (1967). Negationless Intuitionistic Mathematics (Thesis). Minichiello, J. Kent (1969). "An extension of negationless logic" (PDF). Notre Dame Journal of Formal Logic. 10 (3): 298–302. doi:10.1305/ndjfl/1093893719. == External links == Nels David Nelson at the Mathematics Genealogy Project
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Wikipedia:Nelson Merentes#0
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Nelson José Merentes Díaz (born 6 May 1954) is a Venezuelan mathematician, researcher, and politician. == Academic activity == In 1978 Merentes finished his bachelor's degree of Mathematics at Central University of Venezuela and continued his post graduate education taking courses on Economy and Finance, as well as in multifunction techniques for the study of economic problems, completing finally a doctorate in Mathematics with summa cum laude honors, at the Eötvös Loránd University of Budapest (Hungary) (1991). Merentes developed most of his research and teaching at Central University of Venezuela where he participated as professor, representative and member of various councils and committees. == Public office work == Merentes also worked extensively in public administration. From 2000 to 2001 he was the Economy and Finance subcommittee's chairman of the National Legislative Committee. He also worked for the Ministry of Finance as deputy minister of Regulation and Control (2000-2001). In 2001 he was appointed as Minister of Finance of Venezuela by President Hugo Chavez. He held that position until the following year, when he was designated as Science and Technology Minister. From that position he was called by President Chavez for the presidency of Social Development Bank (BANDES), a position he left to return to the Ministry of Finance in early 2004. During his second term, took place the creation of the FONDEN, Venezuela's National Development Fund. From April 2009 he became a president of the Central Bank of Venezuela until 2013. In April 2013 is appointed as Venezuela's Minister of Finance by Nicolás Maduro. In January 2014, he was re-designated as president of the Central Bank of Venezuela. == Sanctions == In 2017, Canada sanctioned Merentes and other Venezuelan officials under the Justice for Victims of Corrupt Foreign Officials Act, stating: "These individuals are responsible for, or complicit in, gross violations of internationally recognized human rights, have committed acts of significant corruption, or both." == Published works == As a researcher Merentes has published more than 200 scientific papers, including some in specialized mathematical study journals. His field of study was mainly focused on the study of differential equations and Lipschitz continuity. some of his notable contributions include: On the Composition Operator in AC[a, b] (1991), On the Composition Operator in BV φ[a; b] (1991) On Functions of Bounded (p,k)-Variation (1992) (with S. Rivas and J. L. Sánchez) Characterization of Globally Lipschitzian Composition Operators in the Banach Space BV2p [a, b] (1992) (with J. Matkowski) Explicit Petree's function of interpolation of the spaces H p s (1993) On the Composition Operator between RVp [a, b] and BV [a, b] (1995) (with S. Rivas) Uniformly Continuous Set-valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Wiener (2010) (with A. Azócar, J. A. Guerrero and J. Matkowski) Locally Lipschitz Composition Operators in Spaces of Functions of Bounded Variation (2010) (with J. Appell y J.L. Sanchez) Measures of Noncompactness in the Study of Asymptotically Stable and Ultimately Nondecreasing Solutions of Integral Equations (2010) (with J. Appell y J. Banaś) Exact Controllability of Semilinear Stochastic Evolution Equation (2011) (with D. Barráez, H. Leiva and M. Narváez) Integral Representation of Functions of Bounded Second Φ-Variation in the Sense of Schramm (2011) (with J. Giménez and S. Rivas) Approximate Controllability of Semilinear Reaction Diffusion Equations (2012) (with H. Leiva and J. L. Sánchez) Uniformly Bounded Set-valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Schramm (2012) (with T. Ereú, J. L. Sánchez and M. Wróbel) == References ==
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Wikipedia:Nerida Ellerton#0
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Nerida Fay Ellerton (née Gersch, born 1942) is an Australian mathematics educator and historian of mathematics. She is professor of mathematics education at Illinois State University. As well as studying the present state of mathematics education, she and her husband McKenzie A. (Ken) Clements have researched the history of mathematics education, in the process discovering school worksheets in the Harvard Library that are among the oldest known writings of Abraham Lincoln. == Education and career == Ellerton was born in 1942; her father was a schoolteacher in a small school in the Australian Outback. She completed a Ph.D. in physical and inorganic chemistry in 1966, at the University of Adelaide; her dissertation was The interaction of aminoacridines and aminobenzacridines with DNA. By the 1980s she worked in mathematics education at Deakin University. She was director of the National Center for Mathematics Education Research from 1992 to 1993, as professor of mathematics education at Edith Cowan University from 1993 to 1997, and as professor and dean of mathematics education at the University of Southern Queensland from 1997 to 2002. While at Edith Cowan University, she also served as editor of the Mathematics Education Research Journal. Her first husband died in 2001, and Ellerton moved to Illinois State University in 2002. In 2005, she married Clements, another Australian mathematics educator and long-term collaborator; he moved to Illinois State to join her. == Books == Ellerton's books include: School Mathematics: The Challenge to Change (edited with M. A. Ken Clements, UNSW Press, 1989) The National Curriculum Debacle (with M. A. Ken Clements, Meridian Press, 1994) Mathematics Education Research: Past, Present, and Future (with M. A. Ken Clements, UNESCO, 1996) Rewriting the History of School Mathematics in North America, 1607–1861: The Central Role of Cyphering Books (with M. A. Ken Clements, Springer, 2012) Abraham Lincoln’s Cyphering Book and Ten other Extraordinary Cyphering Books (with M. A. Ken Clements, Springer, 2014) Thomas Jefferson and his Decimals 1775–1810: Neglected Years in the History of U.S. School Mathematics (with M. A. Ken Clements, Springer, 2015) Mathematical Problem Posing: From Research to Effective Practice (edited with Florence Mihaela Singer and Jinfa Cai, Springer, 2015) Samuel Pepys, Isaac Newton, James Hodgson, and the Beginnings of Secondary School Mathematics: A History of the Royal Mathematical School Within Christ's Hospital, London 1673–1868 (with M. A. Ken Clements, Springer, 2017) Using Design Research and History to Tackle a Fundamental Problem with School Algebra (with M. A. Ken Clements and Sinan Kanbir, Springer, 2017) == References ==
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Wikipedia:Nested radical#0
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In algebra, a nested radical is a radical expression (one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include 5 − 2 5 , {\displaystyle {\sqrt {5-2{\sqrt {5}}\ }},} which arises in discussing the regular pentagon, and more complicated ones such as 2 + 3 + 4 3 3 . {\displaystyle {\sqrt[{3}]{2+{\sqrt {3}}+{\sqrt[{3}]{4}}\ }}.} == Denesting == Some nested radicals can be rewritten in a form that is not nested. For example, 3 + 2 2 = 1 + 2 , {\displaystyle {\sqrt {3+2{\sqrt {2}}}}=1+{\sqrt {2}}\,,} 2 3 − 1 3 = 1 − 2 3 + 4 3 9 3 . {\displaystyle {\sqrt[{3}]{{\sqrt[{3}]{2}}-1}}={\frac {1-{\sqrt[{3}]{2}}+{\sqrt[{3}]{4}}}{\sqrt[{3}]{9}}}\,.} Another simple example, 2 3 = 2 6 {\displaystyle {\sqrt[{3}]{\sqrt {2}}}={\sqrt[{6}]{2}}} Rewriting a nested radical in this way is called denesting. This is not always possible, and, even when possible, it is often difficult. == Two nested square roots == In the case of two nested square roots, the following theorem completely solves the problem of denesting. If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that a + c = x ± y {\displaystyle {\sqrt {a+{\sqrt {c}}}}={\sqrt {x}}\pm {\sqrt {y}}} if and only if a 2 − c {\displaystyle a^{2}-c~} is the square of a rational number d. If the nested radical is real, x and y are the two numbers a + d 2 {\displaystyle {\frac {a+d}{2}}~} and a − d 2 , {\displaystyle ~{\frac {a-d}{2}}~,~} where d = a 2 − c {\displaystyle ~d={\sqrt {a^{2}-c}}~} is a rational number. In particular, if a and c are integers, then 2x and 2y are integers. This result includes denestings of the form a + c = z ± y , {\displaystyle {\sqrt {a+{\sqrt {c}}}}=z\pm {\sqrt {y}}~,} as z may always be written z = ± z 2 , {\displaystyle z=\pm {\sqrt {z^{2}}},} and at least one of the terms must be positive (because the left-hand side of the equation is positive). A more general denesting formula could have the form a + c = α + β x + γ y + δ x y . {\displaystyle {\sqrt {a+{\sqrt {c}}}}=\alpha +\beta {\sqrt {x}}+\gamma {\sqrt {y}}+\delta {\sqrt {x}}{\sqrt {y}}~.} However, Galois theory implies that either the left-hand side belongs to Q ( c ) , {\displaystyle \mathbb {Q} ({\sqrt {c}}),} or it must be obtained by changing the sign of either x , {\displaystyle {\sqrt {x}},} y , {\displaystyle {\sqrt {y}},} or both. In the first case, this means that one can take x = c and γ = δ = 0. {\displaystyle \gamma =\delta =0.} In the second case, α {\displaystyle \alpha } and another coefficient must be zero. If β = 0 , {\displaystyle \beta =0,} one may rename xy as x for getting δ = 0. {\displaystyle \delta =0.} Proceeding similarly if α = 0 , {\displaystyle \alpha =0,} it results that one can suppose α = δ = 0. {\displaystyle \alpha =\delta =0.} This shows that the apparently more general denesting can always be reduced to the above one. Proof: By squaring, the equation a + c = x ± y {\displaystyle {\sqrt {a+{\sqrt {c}}}}={\sqrt {x}}\pm {\sqrt {y}}} is equivalent with a + c = x + y ± 2 x y , {\displaystyle a+{\sqrt {c}}=x+y\pm 2{\sqrt {xy}},} and, in the case of a minus in the right-hand side, (square roots are nonnegative by definition of the notation). As the inequality may always be satisfied by possibly exchanging x and y, solving the first equation in x and y is equivalent with solving a + c = x + y ± 2 x y . {\displaystyle a+{\sqrt {c}}=x+y\pm 2{\sqrt {xy}}.} This equality implies that x y {\displaystyle {\sqrt {xy}}} belongs to the quadratic field Q ( c ) . {\displaystyle \mathbb {Q} ({\sqrt {c}}).} In this field every element may be uniquely written α + β c , {\displaystyle \alpha +\beta {\sqrt {c}},} with α {\displaystyle \alpha } and β {\displaystyle \beta } being rational numbers. This implies that ± 2 x y {\displaystyle \pm 2{\sqrt {xy}}} is not rational (otherwise the right-hand side of the equation would be rational; but the left-hand side is irrational). As x and y must be rational, the square of ± 2 x y {\displaystyle \pm 2{\sqrt {xy}}} must be rational. This implies that α = 0 {\displaystyle \alpha =0} in the expression of ± 2 x y {\displaystyle \pm 2{\sqrt {xy}}} as α + β c . {\displaystyle \alpha +\beta {\sqrt {c}}.} Thus a + c = x + y + β c {\displaystyle a+{\sqrt {c}}=x+y+\beta {\sqrt {c}}} for some rational number β . {\displaystyle \beta .} The uniqueness of the decomposition over 1 and c {\displaystyle {\sqrt {c}}} implies thus that the considered equation is equivalent with a = x + y and ± 2 x y = c . {\displaystyle a=x+y\quad {\text{and}}\quad \pm 2{\sqrt {xy}}={\sqrt {c}}.} It follows by Vieta's formulas that x and y must be roots of the quadratic equation z 2 − a z + c 4 = 0 ; {\displaystyle z^{2}-az+{\frac {c}{4}}=0~;} its Δ = a 2 − c = d 2 > 0 {\displaystyle ~\Delta =a^{2}-c=d^{2}>0~} (≠ 0, otherwise c would be the square of a), hence x and y must be a + a 2 − c 2 {\displaystyle {\frac {a+{\sqrt {a^{2}-c}}}{2}}~} and a − a 2 − c 2 . {\displaystyle ~{\frac {a-{\sqrt {a^{2}-c}}}{2}}~.} Thus x and y are rational if and only if d = a 2 − c {\displaystyle d={\sqrt {a^{2}-c}}~} is a rational number. For explicitly choosing the various signs, one must consider only positive real square roots, and thus assuming c > 0. The equation a 2 = c + d 2 {\displaystyle a^{2}=c+d^{2}} shows that |a| > √c. Thus, if the nested radical is real, and if denesting is possible, then a > 0. Then the solution is a + c = a + d 2 + a − d 2 , a − c = a + d 2 − a − d 2 . {\displaystyle {\begin{aligned}{\sqrt {a+{\sqrt {c}}}}&={\sqrt {\frac {a+d}{2}}}+{\sqrt {\frac {a-d}{2}}},\\[6pt]{\sqrt {a-{\sqrt {c}}}}&={\sqrt {\frac {a+d}{2}}}-{\sqrt {\frac {a-d}{2}}}.\end{aligned}}} == Some identities of Ramanujan == Srinivasa Ramanujan demonstrated a number of curious identities involving nested radicals. Among them are the following: 3 + 2 5 4 3 − 2 5 4 4 = 5 4 + 1 5 4 − 1 = 1 2 ( 3 + 5 4 + 5 + 125 4 ) , {\displaystyle {\sqrt[{4}]{\frac {3+2{\sqrt[{4}]{5}}}{3-2{\sqrt[{4}]{5}}}}}={\frac {{\sqrt[{4}]{5}}+1}{{\sqrt[{4}]{5}}-1}}={\tfrac {1}{2}}\left(3+{\sqrt[{4}]{5}}+{\sqrt {5}}+{\sqrt[{4}]{125}}\right),} 28 3 − 27 3 = 1 3 ( 98 3 − 28 3 − 1 ) , {\displaystyle {\sqrt {{\sqrt[{3}]{28}}-{\sqrt[{3}]{27}}}}={\tfrac {1}{3}}\left({\sqrt[{3}]{98}}-{\sqrt[{3}]{28}}-1\right),} 32 5 5 − 27 5 5 3 = 1 25 5 + 3 25 5 − 9 25 5 , {\displaystyle {\sqrt[{3}]{{\sqrt[{5}]{\frac {32}{5}}}-{\sqrt[{5}]{\frac {27}{5}}}}}={\sqrt[{5}]{\frac {1}{25}}}+{\sqrt[{5}]{\frac {3}{25}}}-{\sqrt[{5}]{\frac {9}{25}}},} and == Landau's algorithm == In 1989 Susan Landau introduced the first algorithm for deciding which nested radicals can be denested. Earlier algorithms worked in some cases but not others. Landau's algorithm involves complex roots of unity and runs in exponential time with respect to the depth of the nested radical. == In trigonometry == In trigonometry, the sines and cosines of many angles can be expressed in terms of nested radicals. For example, sin π 60 = sin 3 ∘ = 1 16 [ 2 ( 1 − 3 ) 5 + 5 + 2 ( 5 − 1 ) ( 3 + 1 ) ] {\displaystyle \sin {\frac {\pi }{60}}=\sin 3^{\circ }={\frac {1}{16}}\left[2(1-{\sqrt {3}}){\sqrt {5+{\sqrt {5}}}}+{\sqrt {2}}({\sqrt {5}}-1)({\sqrt {3}}+1)\right]} and sin π 24 = sin 7.5 ∘ = 1 2 2 − 2 + 3 = 1 2 2 − 1 + 3 2 . {\displaystyle \sin {\frac {\pi }{24}}=\sin 7.5^{\circ }={\frac {1}{2}}{\sqrt {2-{\sqrt {2+{\sqrt {3}}}}}}={\frac {1}{2}}{\sqrt {2-{\frac {1+{\sqrt {3}}}{\sqrt {2}}}}}.} The last equality results directly from the results of § Two nested square roots. == In the solution of the cubic equation == Nested radicals appear in the algebraic solution of the cubic equation. Any cubic equation can be written in simplified form without a quadratic term, as x 3 + p x + q = 0 , {\displaystyle x^{3}+px+q=0,} whose general solution for one of the roots is x = − q 2 + q 2 4 + p 3 27 3 + − q 2 − q 2 4 + p 3 27 3 . {\displaystyle x={\sqrt[{3}]{-{q \over 2}+{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}+{\sqrt[{3}]{-{q \over 2}-{\sqrt {{q^{2} \over 4}+{p^{3} \over 27}}}}}.} In the case in which the cubic has only one real root, the real root is given by this expression with the radicands of the cube roots being real and with the cube roots being the real cube roots. In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to be any specific complex cube root of the complex radicand, and by defining the second cube root to be the complex conjugate of the first one. The nested radicals in this solution cannot in general be simplified unless the cubic equation has at least one rational solution. Indeed, if the cubic has three irrational but real solutions, we have the casus irreducibilis, in which all three real solutions are written in terms of cube roots of complex numbers. On the other hand, consider the equation x 3 − 7 x + 6 = 0 , {\displaystyle x^{3}-7x+6=0,} which has the rational solutions 1, 2, and −3. The general solution formula given above gives the solutions x = − 3 + 10 3 i 9 3 + − 3 − 10 3 i 9 3 . {\displaystyle x={\sqrt[{3}]{-3+{\frac {10{\sqrt {3}}i}{9}}}}+{\sqrt[{3}]{-3-{\frac {10{\sqrt {3}}i}{9}}}}.} For any given choice of cube root and its conjugate, this contains nested radicals involving complex numbers, yet it is reducible (even though not obviously so) to one of the solutions 1, 2, or −3. == Infinitely nested radicals == === Square roots === Under certain conditions infinitely nested square roots such as x = 2 + 2 + 2 + 2 + ⋯ {\displaystyle x={\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots }}}}}}}}} represent rational numbers. This rational number can be found by realizing that x also appears under the radical sign, which gives the equation x = 2 + x . {\displaystyle x={\sqrt {2+x}}.} If we solve this equation, we find that x = 2 (the second solution x = −1 doesn't apply, under the convention that the positive square root is meant). This approach can also be used to show that generally, if n > 0, then n + n + n + n + ⋯ = 1 2 ( 1 + 1 + 4 n ) {\displaystyle {\sqrt {n+{\sqrt {n+{\sqrt {n+{\sqrt {n+\cdots }}}}}}}}={\tfrac {1}{2}}\left(1+{\sqrt {1+4n}}\right)} and is the positive root of the equation x2 − x − n = 0. For n = 1, this root is the golden ratio φ, approximately equal to 1.618. The same procedure also works to obtain, if n > 0, n − n − n − n − ⋯ = 1 2 ( − 1 + 1 + 4 n ) , {\displaystyle {\sqrt {n-{\sqrt {n-{\sqrt {n-{\sqrt {n-\cdots }}}}}}}}={\tfrac {1}{2}}\left(-1+{\sqrt {1+4n}}\right),} which is the positive root of the equation x2 + x − n = 0. ==== Nested square roots of 2 ==== The nested square roots of 2 are a special case of the wide class of infinitely nested radicals. There are many known results that bind them to sines and cosines. For example, it has been shown that nested square roots of 2 as R ( b k , … , b 1 ) = b k 2 2 + b k − 1 2 + b k − 2 2 + ⋯ + b 2 2 + x {\displaystyle R(b_{k},\ldots ,b_{1})={\frac {b_{k}}{2}}{\sqrt {2+b_{k-1}{\sqrt {2+b_{k-2}{\sqrt {2+\cdots +b_{2}{\sqrt {2+x}}}}}}}}} where x = 2 sin ( π b 1 / 4 ) {\displaystyle x=2\sin(\pi b_{1}/4)} with b 1 {\displaystyle b_{1}} in [−2,2] and b i ∈ { − 1 , 0 , 1 } {\displaystyle b_{i}\in \{-1,0,1\}} for i ≠ 1 {\displaystyle i\neq 1} , are such that R ( b k , … , b 1 ) = cos θ {\displaystyle R(b_{k},\ldots ,b_{1})=\cos \theta } for θ = ( 1 2 − b k 4 − b k b k − 1 8 − b k b k − 1 b k − 2 16 − ⋯ − b k b k − 1 ⋯ b 1 2 k + 1 ) π . {\displaystyle \theta =\left({\frac {1}{2}}-{\frac {b_{k}}{4}}-{\frac {b_{k}b_{k-1}}{8}}-{\frac {b_{k}b_{k-1}b_{k-2}}{16}}-\cdots -{\frac {b_{k}b_{k-1}\cdots b_{1}}{2^{k+1}}}\right)\pi .} This result allows to deduce for any x ∈ [ − 2 , 2 ] {\displaystyle x\in [-2,2]} the value of the following infinitely nested radicals consisting of k nested roots as R k ( x ) = 2 + 2 + ⋯ + 2 + x . {\displaystyle R_{k}(x)={\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2+x}}}}}}.} If x ≥ 2 {\displaystyle x\geq 2} , then R k ( x ) = 2 + 2 + ⋯ + 2 + x = ( x + x 2 − 4 2 ) 1 / 2 k + ( x + x 2 − 4 2 ) − 1 / 2 k {\displaystyle {\begin{aligned}R_{k}(x)&={\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2+x}}}}}}\\&=\left({\frac {x+{\sqrt {x^{2}-4}}}{2}}\right)^{1/2^{k}}+\left({\frac {x+{\sqrt {x^{2}-4}}}{2}}\right)^{-1/2^{k}}\end{aligned}}} These results can be used to obtain some nested square roots representations of π {\displaystyle \pi } . Let us consider the term R ( b k , … , b 1 ) {\displaystyle R\left(b_{k},\ldots ,b_{1}\right)} defined above. Then π = lim k → ∞ [ 2 k + 1 2 − b 1 R ( 1 , − 1 , 1 , 1 , … , 1 , 1 , b 1 ⏟ k terms ) ] {\displaystyle \pi =\lim _{k\rightarrow \infty }\left[{\frac {2^{k+1}}{2-b_{1}}}R(\underbrace {1,-1,1,1,\ldots ,1,1,b_{1}} _{k{\text{ terms }}})\right]} where b 1 ≠ 2 {\displaystyle b_{1}\neq 2} . ==== Ramanujan's infinite radicals ==== Ramanujan posed the following problem to the Journal of Indian Mathematical Society: ? = 1 + 2 1 + 3 1 + ⋯ . {\displaystyle ?={\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}.} This can be solved by noting a more general formulation: ? = a x + ( n + a ) 2 + x a ( x + n ) + ( n + a ) 2 + ( x + n ) ⋯ . {\displaystyle ?={\sqrt {ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\mathrm {\cdots } }}}}}}.} Setting this to F(x) and squaring both sides gives us F ( x ) 2 = a x + ( n + a ) 2 + x a ( x + n ) + ( n + a ) 2 + ( x + n ) ⋯ , {\displaystyle F(x)^{2}=ax+(n+a)^{2}+x{\sqrt {a(x+n)+(n+a)^{2}+(x+n){\sqrt {\mathrm {\cdots } }}}},} which can be simplified to F ( x ) 2 = a x + ( n + a ) 2 + x F ( x + n ) . {\displaystyle F(x)^{2}=ax+(n+a)^{2}+xF(x+n).} It can be shown that F ( x ) = x + n + a {\displaystyle F(x)={x+n+a}} satisfies the equation for F ( x ) {\displaystyle F(x)} , so it can be hoped that it is the true solution. For a complete proof, we would need to show that this is indeed the solution to the equation for F ( x ) {\displaystyle F(x)} . So, setting a = 0, n = 1, and x = 2, we have 3 = 1 + 2 1 + 3 1 + ⋯ . {\displaystyle 3={\sqrt {1+2{\sqrt {1+3{\sqrt {1+\cdots }}}}}}.} Ramanujan stated the following infinite radical denesting in his lost notebook: 5 + 5 + 5 − 5 + 5 + 5 + 5 − ⋯ = 2 + 5 + 15 − 6 5 2 . {\displaystyle {\sqrt {5+{\sqrt {5+{\sqrt {5-{\sqrt {5+{\sqrt {5+{\sqrt {5+{\sqrt {5-\cdots }}}}}}}}}}}}}}={\frac {2+{\sqrt {5}}+{\sqrt {15-6{\sqrt {5}}}}}{2}}.} The repeating pattern of the signs is ( + , + , − , + ) . {\displaystyle (+,+,-,+).} ==== Viète's expression for π ==== Viète's formula for π, the ratio of a circle's circumference to its diameter, is 2 π = 2 2 ⋅ 2 + 2 2 ⋅ 2 + 2 + 2 2 ⋯ . {\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots .} === Cube roots === In certain cases, infinitely nested cube roots such as x = 6 + 6 + 6 + 6 + ⋯ 3 3 3 3 {\displaystyle x={\sqrt[{3}]{6+{\sqrt[{3}]{6+{\sqrt[{3}]{6+{\sqrt[{3}]{6+\cdots }}}}}}}}} can represent rational numbers as well. Again, by realizing that the whole expression appears inside itself, we are left with the equation x = 6 + x 3 . {\displaystyle x={\sqrt[{3}]{6+x}}.} If we solve this equation, we find that x = 2. More generally, we find that n + n + n + n + ⋯ 3 3 3 3 {\displaystyle {\sqrt[{3}]{n+{\sqrt[{3}]{n+{\sqrt[{3}]{n+{\sqrt[{3}]{n+\cdots }}}}}}}}} is the positive real root of the equation x3 − x − n = 0 for all n > 0. For n = 1, this root is the plastic ratio ρ, approximately equal to 1.3247. The same procedure also works to get n − n − n − n − ⋯ 3 3 3 3 {\displaystyle {\sqrt[{3}]{n-{\sqrt[{3}]{n-{\sqrt[{3}]{n-{\sqrt[{3}]{n-\cdots }}}}}}}}} as the real root of the equation x3 + x − n = 0 for all n > 1. === Herschfeld's convergence theorem === An infinitely nested radical a 1 + a 2 + ⋯ {\displaystyle {\sqrt {a_{1}+{\sqrt {a_{2}+\dotsb }}}}} (where all a i {\displaystyle a_{i}} are nonnegative) converges if and only if there is some M ∈ R {\displaystyle M\in \mathbb {R} } such that M ≥ a n 2 − n {\displaystyle M\geq a_{n}^{2^{-n}}} for all n {\displaystyle n} , or in other words sup a n 2 − n < + ∞ . {\textstyle \sup a_{n}^{2^{-n}}<+\infty .} ==== Proof of "if" ==== We observe that a 1 + a 2 + ⋯ ≤ M 2 1 + M 2 2 + ⋯ = M 1 + 1 + ⋯ < 2 M . {\displaystyle {\sqrt {a_{1}+{\sqrt {a_{2}+\dotsb }}}}\leq {\sqrt {M^{2^{1}}+{\sqrt {M^{2^{2}}+\cdots }}}}=M{\sqrt {1+{\sqrt {1+\dotsb }}}}<2M.} Moreover, the sequence ( a 1 + a 2 + … a n ) {\displaystyle \left({\sqrt {a_{1}+{\sqrt {a_{2}+\dotsc {\sqrt {a_{n}}}}}}}\right)} is monotonically increasing. Therefore it converges, by the monotone convergence theorem. ==== Proof of "only if" ==== If the sequence ( a 1 + a 2 + ⋯ a n ) {\displaystyle \left({\sqrt {a_{1}+{\sqrt {a_{2}+\cdots {\sqrt {a_{n}}}}}}}\right)} converges, then it is bounded. However, a n 2 − n ≤ a 1 + a 2 + ⋯ a n {\displaystyle a_{n}^{2^{-n}}\leq {\sqrt {a_{1}+{\sqrt {a_{2}+\cdots {\sqrt {a_{n}}}}}}}} , hence ( a n 2 − n ) {\displaystyle \left(a_{n}^{2^{-n}}\right)} is also bounded. == See also == Exponentiation Sum of radicals == References == === Further reading === Landau, Susan (1994). "How to Tangle with a Nested Radical". Mathematical Intelligencer. 16 (2): 49–55. doi:10.1007/bf03024284. S2CID 119991567. Decreasing the Nesting Depth of Expressions Involving Square Roots Simplifying Square Roots of Square Roots Weisstein, Eric W. "Square Root". MathWorld. Weisstein, Eric W. "Nested Radical". MathWorld.
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Wikipedia:Neusis construction#0
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In geometry, the neusis (νεῦσις; from Ancient Greek νεύειν (neuein) 'incline towards'; plural: νεύσεις, neuseis) is a geometric construction method that was used in antiquity by Greek mathematicians. == Geometric construction == The neusis construction consists of fitting a line element of given length (a) in between two given lines (l and m), in such a way that the line element, or its extension, passes through a given point P. That is, one end of the line element has to lie on l, the other end on m, while the line element is "inclined" towards P. Point P is called the pole of the neusis, line l the directrix, or guiding line, and line m the catch line. Length a is called the diastema (Greek: διάστημα, lit. 'distance'). A neusis construction might be performed by means of a marked ruler that is rotatable around the point P (this may be done by putting a pin into the point P and then pressing the ruler against the pin). In the figure one end of the ruler is marked with a yellow eye with crosshairs: this is the origin of the scale division on the ruler. A second marking on the ruler (the blue eye) indicates the distance a from the origin. The yellow eye is moved along line l, until the blue eye coincides with line m. The position of the line element thus found is shown in the figure as a dark blue bar. If we require that both two marking of the ruler must land on straight line, then the construction is called line-line neusis. Line-circle neusis and circle-cirle neusis are defined in similar way. The line-line neusis gives us precisely the power to solve quadratic and cubic (and hence also quartic) equations while line-circle neusis and circle-circle neuis is strictly more powerful than line-line neusis. Technically, any point generated by either the line-circle neusis or the circle-circle neusis lies in an extension field of the rationals that can be reached by a tower of fields in which each adjacent pair has index either 2, 3, 5, or 6 while the adjacent-pair indices over the tower of the extension field of line-line neusis are either 2 or 3. == Trisection of an angle == Let l be the horizontal line in the adjacent diagram. Angle a (left of point B) is the subject of trisection. First, a point A is drawn at an angle's ray, one unit apart from B. A circle of radius AB is drawn. Then, the markedness of the ruler comes into play: one mark of the ruler is placed at A and the other at B. While keeping the ruler (but not the mark) touching A, the ruler is slid and rotated until one mark is on the circle and the other is on the line l. The mark on the circle is labeled C and the mark on the line is labeled D. Angle b = CDB is equal to one-third of angle a. == Use of the neusis == Neuseis have been important because they sometimes provide a means to solve geometric problems that are not solvable by means of compass and straightedge alone. Examples are the trisection of any angle in three equal parts, and the doubling of the cube. Mathematicians such as Archimedes of Syracuse (287–212 BC) and Pappus of Alexandria (290–350 AD) freely used neuseis; Sir Isaac Newton (1642–1726) followed their line of thought, and also used neusis constructions. Nevertheless, gradually the technique dropped out of use. === Regular polygons === In 2002, A. Baragar showed that every point constructible with marked ruler and compass lies in a tower of fields over Q {\displaystyle \mathbb {Q} } , Q = K 0 ⊂ K 1 ⊂ ⋯ ⊂ K n = K {\displaystyle \mathbb {Q} =K_{0}\subset K_{1}\subset \dots \subset K_{n}=K} , such that the degree of the extension at each step is no higher than 6. Of all prime-power polygons below the 128-gon, this is enough to show that the regular 23-, 29-, 43-, 47-, 49-, 53-, 59-, 67-, 71-, 79-, 83-, 89-, 103-, 107-, 113-, 121-, and 127-gons cannot be constructed with neusis. (If a regular p-gon is constructible, then ζ p = e 2 π i p {\displaystyle \zeta _{p}=e^{\frac {2\pi i}{p}}} is constructible, and in these cases p − 1 has a prime factor higher than 5.) The 3-, 4-, 5-, 6-, 8-, 10-, 12-, 15-, 16-, 17-, 20-, 24-, 30-, 32-, 34-, 40-, 48-, 51-, 60-, 64-, 68-, 80-, 85-, 96-, 102-, 120-, and 128-gons can be constructed with only a straightedge and compass, and the 7-, 9-, 13-, 14-, 18-, 19-, 21-, 26-, 27-, 28-, 35-, 36-, 37-, 38-, 39-, 42-, 52-, 54-, 56-, 57-, 63-, 65-, 70-, 72-, 73-, 74-, 76-, 78-, 81-, 84-, 91-, 95-, 97-, 104-, 105-, 108-, 109-, 111-, 112-, 114-, 117-, 119-, and 126-gons with angle trisection. However, it is not known in general if all quintics (fifth-order polynomials) have neusis-constructible roots, which is relevant for the 11-, 25-, 31-, 41-, 61-, 101-, and 125-gons. Benjamin and Snyder showed in 2014 that the regular 11-gon is neusis-constructible; the 25-, 31-, 41-, 61-, 101-, and 125-gons remain open problems. More generally, the constructibility of all powers of 5 greater than 5 itself by marked ruler and compass is an open problem, along with all primes greater than 11 of the form p = 2r3s5t + 1 where t > 0 (all prime numbers that are greater than 11 and equal to one more than a regular number that is divisible by 10). == Waning popularity == T. L. Heath, the historian of mathematics, has suggested that the Greek mathematician Oenopides (c. 440 BC) was the first to put compass-and-straightedge constructions above neuseis. The principle to avoid neuseis whenever possible may have been spread by Hippocrates of Chios (c. 430 BC), who originated from the same island as Oenopides, and who was—as far as we know—the first to write a systematically ordered geometry textbook. One hundred years after him Euclid too shunned neuseis in his very influential textbook, The Elements. The next attack on the neusis came when, from the fourth century BC, Plato's idealism gained ground. Under its influence a hierarchy of three classes of geometrical constructions was developed. Descending from the "abstract and noble" to the "mechanical and earthly", the three classes were: constructions with straight lines and circles only (compass and straightedge); constructions that in addition to this use conic sections (ellipses, parabolas, hyperbolas); constructions that needed yet other means of construction, for example neuseis. In the end the use of neusis was deemed acceptable only when the two other, higher categories of constructions did not offer a solution. Neusis became a kind of last resort that was invoked only when all other, more respectable, methods had failed. Using neusis where other construction methods might have been used was branded by the late Greek mathematician Pappus of Alexandria (c. 325 AD) as "a not inconsiderable error". == See also == Angle trisection Constructible polygon Pierpont prime Quadratrix Steel square Tomahawk (geometry) Trisectrix == References == R. Boeker, 'Neusis', in: Paulys Realencyclopädie der Classischen Altertumswissenschaft, G. Wissowa red. (1894–), Supplement 9 (1962) 415–461.–In German. The most comprehensive survey; however, the author sometimes has rather curious opinions. T. L. Heath, A history of Greek Mathematics (2 volumes; Oxford 1921). H. G. Zeuthen, Die Lehre von den Kegelschnitten im Altertum [= The Theory of Conic Sections in Antiquity] (Copenhagen 1886; reprinted Hildesheim 1966). == External links == MathWorld page Angle Trisection by Paper Folding
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Wikipedia:Neutral density#0
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In photography and optics, a neutral-density filter, or ND filter, is a filter that reduces or modifies the intensity of all wavelengths, or colors, of light equally, giving no changes in hue of color rendition. It can be a colorless (clear) or grey filter, and is denoted by Wratten number 96. The purpose of a standard photographic neutral-density filter is to reduce the amount of light entering the lens. Doing so allows the photographer to select combinations of aperture, exposure time and sensor sensitivity that would otherwise produce overexposed pictures. This is done to achieve effects such as a shallower depth of field or motion blur of a subject in a wider range of situations and atmospheric conditions. For example, one might wish to photograph a waterfall at a slow shutter speed to create a deliberate motion-blur effect. The photographer might determine that to obtain the desired effect, a shutter speed of ten seconds was needed. On a very bright day, there might be so much light that even at minimal film speed and a minimal aperture, the ten-second shutter speed would let in too much light, and the photo would be overexposed. In this situation, applying an appropriate neutral-density filter is the equivalent of stopping down one or more additional stops, allowing the slower shutter speed and the desired motion-blur effect. == Mechanism == the term "Neutral-density filter" refers to a filter that blocks a range of wavelengths evenly, mechanisms and constructions vary. Reflective ND filters use thin coatings, reflecting light back to the source. The coating varies and can be specialized for use-case and spectrum, often consisting of metal ions. Absorptive filters change the composition of the glass itself and include an anti-reflective coating. These are used in more specialty applications like micro-photography. For an ND filter with optical density d, the fraction of the optical power transmitted through the filter can be calculated as Fractional transmittance ≡ I I 0 = 10 − d , {\displaystyle {\text{Fractional transmittance}}\equiv {\frac {I}{I_{0}}}=10^{-d},} where I is the intensity after the filter, and I0 is the incident intensity. == Uses == The use of an ND filter allows the photographer to use a larger aperture that is at or below the diffraction limit, which varies depending on the size of the sensory medium (film or digital) and for many cameras is between f/8 and f/11, with smaller sensory medium sizes needing larger-sized apertures, and larger ones able to use smaller apertures. ND filters can also be used to reduce the depth of field of an image (by allowing the use of a larger aperture) where otherwise not possible due to a maximal shutter speed limit. Instead of reducing the aperture to limit light, the photographer can add a ND filter to limit light, and can then set the shutter speed according to the particular motion desired (blur of water movement, for example) and the aperture set as needed (small aperture for maximal sharpness or large aperture for narrow depth of field (subject in focus and background out of focus)). Using a digital camera, the photographer can see the image right away and choose the best ND filter to use for the scene being captured by first knowing the best aperture to use for maximal sharpness desired. The shutter speed would be selected by finding the desired blur from subject movement. The camera would be set up for these in manual mode, and then the overall exposure adjusted darker by adjusting either aperture or shutter speed, noting the number of stops needed to bring the exposure to that which is desired. That offset would then be the amount of stops needed in the ND filter to use for that scene. Examples of this use include: Blurring water motion (e.g. waterfalls, rivers, oceans). Reducing depth of field in very bright light (e.g. daylight). When using a flash on a camera with a focal-plane shutter, exposure time is limited to the maximal speed (often 1/250th of a second, at best), at which the entire film or sensor is exposed to light at one instant. Without an ND filter, this can result in the need to use f/8 or higher. Using a wider aperture to stay below the diffraction limit. Reduce the visibility of moving objects. Add motion blur to subjects. Extended time exposures Neutral-density filters are used to control exposure with photographic catadioptric lenses, since the use of a traditional iris diaphragm increases the ratio of the central obstruction found in those systems, leading to poor performance. ND filters find applications in several high-precision laser experiments because the power of a laser cannot be adjusted without changing other properties of the laser light (e.g. collimation of the beam). Moreover, most lasers have a minimal power setting at which they can be operated. To achieve the desired light attenuation, one or more neutral-density filters can be placed in the path of the beam. Large telescopes can cause the Moon and planets to become too bright and lose contrast. A neutral-density filter can increase the contrast and cut down the brightness, making these objects easier to view. == Varieties == A graduated ND filter is similar, except that the intensity varies across the surface of the filter. This is useful when one region of the image is bright and the rest is not, as in a picture of a sunset. The transition area, or edge, is available in different variations (soft, hard, attenuator). The most common is a soft edge and provides a smooth transition from the ND side and the clear side. Hard-edge filters have a sharp transition from ND to clear, and the attenuator edge changes gradually over most of the filter, so the transition is less noticeable. Another type of ND filter configuration is the ND-filter wheel. It consists of two perforated glass disks that have progressively denser coating applied around the perforation on the face of each disk. When the two disks are counter-rotated in front of each other, they gradually and evenly go from 100% transmission to 0% transmission. These are used on catadioptric telescopes mentioned above and in any system that is required to work at 100% of its aperture (usually because the system is required to work at its maximal angular resolution). In practice, ND filters are not perfect, as they do not reduce the intensity of all wavelengths equally. This can sometimes create color casts in recorded images, particularly with inexpensive filters. More significantly, most ND filters are only specified over the visible region of the spectrum and do not proportionally block all wavelengths of ultraviolet or infrared radiation. This can be dangerous if using ND filters to view sources (such as the Sun or white-hot metal or glass), which emit intense invisible radiation, since the eye may be damaged even though the source does not look bright when viewed through the filter. Special filters must be used if such sources are to be safely viewed. An inexpensive, homemade alternative to professional ND filters can be made from a piece of welder's glass. Depending on the rating of the welder's glass, this can have the effect of a 10-stop filter. === Variable neutral-density filter === One main disadvantage of neutral-density (ND) filters is that different shooting situations often require a variety of filters, which can become quite expensive. For example, using screw-on filters requires a separate set for each lens diameter, though inexpensive step-up rings can minimize this requirement. To address this issue, some manufacturers have developed variable ND filters. These filters consist of two polarizing filters, with at least one being rotatable. The rear filter blocks light in one plane, while the front filter can be adjusted. As the front filter rotates, it cuts down the amount of light reaching the camera sensor, allowing for nearly infinite control over light levels. The advantage of this approach is reduced bulk and expenses, but one drawback is a loss of image quality caused by both using two elements together and by combining two polarizing filters. === Extreme ND filters === To create ethereal looking landscapes and seascapes with extremely blurred water or other motion, the use of multiple stacked ND filters might be required. This has, as in the case of variable NDs, the effect of reducing image quality. To counter this, some manufacturers have produced high-quality extreme ND filters. Typically these are rated at a 10-stop reduction, allowing very slow shutter speeds even in relatively bright conditions. == Ratings == In photography, ND filters are quantified by their optical density or equivalently their f-stop reduction. In microscopy, the transmittance value is sometimes used. In astronomy, the fractional transmittance is sometimes used (eclipses). Note: Hoya, B+W, Cokin use code ND2 or ND2x, etc.; Lee, Tiffen use code 0.3ND, etc.; Leica uses code 1×, 4×, 8×, etc. Note: ND 3.8 is the correct value for solar CCD exposure without risk of electronic damage. Note: ND 5.0 is the minimum for direct eye solar observation without damage of retina. A further check must be performed for the particular filter used, checking on the spectrogram that also UV and IR are mitigated with the same value. Note: ANSI shades are defined by standard as ranges with central values. They are here approximated using the equation from ANSI Z87.1, S = 7 3 l o g 10 1 T L + 1 {\displaystyle S={\dfrac {7}{3}}log_{10}{\dfrac {1}{T_{L}}}+1} , which bases luminous transmittance ( T L {\displaystyle T_{L}} ) on CIE Illuminant A; ANSI shade numbers have much lower limits for ultraviolet transmittance. == References == == External links == Neutral Density Filter Calculation Chart Neutral Density Filters and Graduated ND Filters What do ND filters do, and what do they NOT do? Neutral Density Filters: What are they & when to use them ? Archived 2017-06-13 at the Wayback Machine Neutral Density Filter FAQ at Digital Grin Photography Forum
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Wikipedia:New York Number Theory Seminar#0
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The New York Number Theory Seminar is a research seminar devoted to the theory of numbers and related parts of mathematics and physics. The seminar began in 1982 under the founding organizers Harvey Cohn, David and Gregory Chudnovsky, and Melvyn B. Nathanson. It is held at the Graduate Center, CUNY. == Overview == The New York Number Theory Seminar began in January 1982 and was originally organized by number theorists Harvey Cohn, David and Gregory Chudnovsky, and Melvyn B. Nathanson. Since the retirement of Cohn, Nathanson is the sole organizer. The seminar also organizes an annual Workshop on Combinatorial and Additive Number Theory (CANT) at the Graduate Center, CUNY. == Publications == Four volumes of the collected lecture notes of the seminar were published in the Lecture Notes in Mathematics series by Springer-Verlag. These volumes covered the seminar from 1982 to 1988. Three additional stand-alone books were published by Springer-Verlag under the title Number Theory, covering the seminar between 1989 and 2003. == External links == Official website == References ==
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Wikipedia:Newton fractal#0
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The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial p(z) ∈ C {\displaystyle \mathbb {C} } [z] or transcendental function. It is the Julia set of the meromorphic function z ↦ z − p(z)/p′(z) which is given by Newton's method. When there are no attractive cycles (of order greater than 1), it divides the complex plane into regions Gk, each of which is associated with a root ζk of the polynomial, k = 1, …, deg(p). In this way the Newton fractal is similar to the Mandelbrot set, and like other fractals it exhibits an intricate appearance arising from a simple description. It is relevant to numerical analysis because it shows that (outside the region of quadratic convergence) the Newton method can be very sensitive to its choice of start point. Almost all points of the complex plane are associated with one of the deg(p) roots of a given polynomial in the following way: the point is used as starting value z0 for Newton's iteration zn + 1 := zn − p(zn)/p'(zn), yielding a sequence of points z1, z2, …, If the sequence converges to the root ζk, then z0 was an element of the region Gk. However, for every polynomial of degree at least 2 there are points for which the Newton iteration does not converge to any root: examples are the boundaries of the basins of attraction of the various roots. There are even polynomials for which open sets of starting points fail to converge to any root: a simple example is z3 − 2z + 2, where some points are attracted by the cycle 0, 1, 0, 1… rather than by a root. An open set for which the iterations converge towards a given root or cycle (that is not a fixed point), is a Fatou set for the iteration. The complementary set to the union of all these, is the Julia set. The Fatou sets have common boundary, namely the Julia set. Therefore, each point of the Julia set is a point of accumulation for each of the Fatou sets. It is this property that causes the fractal structure of the Julia set (when the degree of the polynomial is larger than 2). To plot images of the fractal, one may first choose a specified number d of complex points (ζ1, …, ζd) and compute the coefficients (p1, …, pd) of the polynomial p ( z ) = z d + p 1 z d − 1 + ⋯ + p d − 1 z + p d := ( z − ζ 1 ) ( z − ζ 2 ) ⋯ ( z − ζ d ) {\displaystyle p(z)=z^{d}+p_{1}z^{d-1}+\cdots +p_{d-1}z+p_{d}:=(z-\zeta _{1})(z-\zeta _{2})\cdots (z-\zeta _{d})} . Then for a rectangular lattice z m n = z 00 + m Δ x + i n Δ y ; m = 0 , … , M − 1 ; n = 0 , … , N − 1 {\displaystyle z_{mn}=z_{00}+m\,\Delta x+in\,\Delta y;\quad m=0,\ldots ,M-1;\quad n=0,\ldots ,N-1} of points in C {\displaystyle \mathbb {C} } , one finds the index k(m,n) of the corresponding root ζk(m,n) and uses this to fill an M × N raster grid by assigning to each point (m,n) a color fk(m,n). Additionally or alternatively the colors may be dependent on the distance D(m,n), which is defined to be the first value D such that |zD − ζk(m,n)| < ε for some previously fixed small ε > 0. == Generalization of Newton fractals == A generalization of Newton's iteration is z n + 1 = z n − a p ( z n ) p ′ ( z n ) {\displaystyle z_{n+1}=z_{n}-a{\frac {p(z_{n})}{p'(z_{n})}}} where a is any complex number. The special choice a = 1 corresponds to the Newton fractal. The fixed points of this map are stable when a lies inside the disk of radius 1 centered at 1. When a is outside this disk, the fixed points are locally unstable, however the map still exhibits a fractal structure in the sense of Julia set. If p is a polynomial of degree d, then the sequence zn is bounded provided that a is inside a disk of radius d centered at d. More generally, Newton's fractal is a special case of a Julia set. Serie : p(z) = zn- 1 Other fractals where potential and trigonometric functions are multiplied. p(z) = zn*Sin(z) - 1 === Nova fractal === The Nova fractal invented in the mid 1990s by Paul Derbyshire, is a generalization of the Newton fractal with the addition of a value c at each step: z n + 1 = z n − a p ( z n ) p ′ ( z n ) + c = G ( a , c , z ) {\displaystyle z_{n+1}=z_{n}-a{\frac {p(z_{n})}{p'(z_{n})}}+c=G(a,c,z)} The "Julia" variant of the Nova fractal keeps c constant over the image and initializes z0 to the pixel coordinates. The "Mandelbrot" variant of the Nova fractal initializes c to the pixel coordinates and sets z0 to a critical point, where ∂ ∂ z G ( a , c , z ) = 0. {\displaystyle {\frac {\partial }{\partial z}}G(a,c,z)=0.} Commonly-used polynomials like p(z) = z3 − 1 or p(z) = (z − 1)3 lead to a critical point at z = 1. == Implementation == In order to implement the Newton fractal, it is necessary to have a starting function as well as its derivative function: f ( z ) = z 3 − 1 f ′ ( z ) = 3 z 2 {\displaystyle {\begin{aligned}f(z)&=z^{3}-1\\f'(z)&=3z^{2}\end{aligned}}} The three roots of the function are z = 1 , − 1 2 + 3 2 i , − 1 2 − 3 2 i {\displaystyle z=1,\ -{\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i,\ -{\tfrac {1}{2}}-{\tfrac {\sqrt {3}}{2}}i} The above-defined functions can be translated in pseudocode as follows: It is now just a matter of implementing the Newton method using the given functions. == See also == Julia set Mandelbrot set == References == == Further reading == J. H. Hubbard, D. Schleicher, S. Sutherland: How to Find All Roots of Complex Polynomials by Newton's Method, Inventiones Mathematicae vol. 146 (2001) – with a discussion of the global structure of Newton fractals On the Number of Iterations for Newton's Method by Dierk Schleicher July 21, 2000 Newton's Method as a Dynamical System by Johannes Rueckert Newton's Fractal (which Newton knew nothing about) by 3Blue1Brown, along with an interactive demonstration of the fractal on his website, and the source code for the demonstration
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Wikipedia:Newton polygon#0
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In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the ultrametric field of interest was essentially the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring K [ [ X ] ] {\displaystyle K[[X]]} , over K {\displaystyle K} , where K {\displaystyle K} was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading terms a X r {\displaystyle aX^{r}} of the power series expansion solutions to equations P ( F ( X ) ) = 0 {\displaystyle P(F(X))=0} where P {\displaystyle P} is a polynomial with coefficients in K [ X ] {\displaystyle K[X]} , the polynomial ring; that is, implicitly defined algebraic functions. The exponents r {\displaystyle r} here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in K [ [ Y ] ] {\displaystyle K[[Y]]} with Y = X 1 d {\displaystyle Y=X^{\frac {1}{d}}} for a denominator d {\displaystyle d} corresponding to the branch. The Newton polygon gives an effective, algorithmic approach to calculating d {\displaystyle d} . After the introduction of the p-adic numbers, it was shown that the Newton polygon is just as useful in questions of ramification for local fields, and hence in algebraic number theory. Newton polygons have also been useful in the study of elliptic curves. == Definition == A priori, given a polynomial over a field, the behaviour of the roots (assuming it has roots) will be unknown. Newton polygons provide one technique for the study of the behaviour of the roots. Let K {\displaystyle K} be a field endowed with a non-archimedean valuation v K : K → R ∪ { ∞ } {\displaystyle v_{K}:K\to \mathbb {R} \cup \{\infty \}} , and let f ( x ) = a n x n + ⋯ + a 1 x + a 0 ∈ K [ x ] , {\displaystyle f(x)=a_{n}x^{n}+\cdots +a_{1}x+a_{0}\in K[x],} with a 0 a n ≠ 0 {\displaystyle a_{0}a_{n}\neq 0} . Then the Newton polygon of f {\displaystyle f} is defined to be the lower boundary of the convex hull of the set of points P i = ( i , v K ( a i ) ) , {\displaystyle P_{i}=\left(i,v_{K}(a_{i})\right),} ignoring the points with a i = 0 {\displaystyle a_{i}=0} . Restated geometrically, plot all of these points Pi on the xy-plane. Let's assume that the points indices increase from left to right (P0 is the leftmost point, Pn is the rightmost point). Then, starting at P0, draw a ray straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point Pk1 (not necessarily P1). Break the ray here. Now draw a second ray from Pk1 straight down parallel with the y-axis, and rotate this ray counter-clockwise until it hits the point Pk2. Continue until the process reaches the point Pn; the resulting polygon (containing the points P0, Pk1, Pk2, ..., Pkm, Pn) is the Newton polygon. Another, perhaps more intuitive way to view this process is this : consider a rubber band surrounding all the points P0, ..., Pn. Stretch the band upwards, such that the band is stuck on its lower side by some of the points (the points act like nails, partially hammered into the xy plane). The vertices of the Newton polygon are exactly those points. For a neat diagram of this see Ch6 §3 of "Local Fields" by JWS Cassels, LMS Student Texts 3, CUP 1986. It is on p99 of the 1986 paperback edition. == Main theorem == With the notations in the previous section, the main result concerning the Newton polygon is the following theorem, which states that the valuation of the roots of f {\displaystyle f} are entirely determined by its Newton polygon: Let μ 1 , μ 2 , … , μ r {\displaystyle \mu _{1},\mu _{2},\ldots ,\mu _{r}} be the slopes of the line segments of the Newton polygon of f ( x ) {\displaystyle f(x)} (as defined above) arranged in increasing order, and let λ 1 , λ 2 , … , λ r {\displaystyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{r}} be the corresponding lengths of the line segments projected onto the x-axis (i.e. if we have a line segment stretching between the points P i {\displaystyle P_{i}} and P j {\displaystyle P_{j}} then the length is j − i {\displaystyle j-i} ). The μ i {\displaystyle \mu _{i}} are distinct; ∑ i λ i = n {\displaystyle \sum _{i}\lambda _{i}=n} ; if α {\displaystyle \alpha } is a root of f {\displaystyle f} in K {\displaystyle K} , v ( α ) ∈ { − μ 1 , … , − μ r } {\displaystyle v(\alpha )\in \{-\mu _{1},\ldots ,-\mu _{r}\}} ; for every i {\displaystyle i} , the number of roots of f {\displaystyle f} whose valuations are equal to − μ i {\displaystyle -\mu _{i}} (counting multiplicities) is at most λ i {\displaystyle \lambda _{i}} , with equality if f {\displaystyle f} splits into the product of linear factors over K {\displaystyle K} . == Corollaries and applications == With the notation of the previous sections, we denote, in what follows, by L {\displaystyle L} the splitting field of f {\displaystyle f} over K {\displaystyle K} , and by v L {\displaystyle v_{L}} an extension of v K {\displaystyle v_{K}} to L {\displaystyle L} . Newton polygon theorem is often used to show the irreducibility of polynomials, as in the next corollary for example: Suppose that the valuation v {\displaystyle v} is discrete and normalized, and that the Newton polynomial of f {\displaystyle f} contains only one segment whose slope is μ {\displaystyle \mu } and projection on the x-axis is λ {\displaystyle \lambda } . If μ = a / n {\displaystyle \mu =a/n} , with a {\displaystyle a} coprime to n {\displaystyle n} , then f {\displaystyle f} is irreducible over K {\displaystyle K} . In particular, since the Newton polygon of an Eisenstein polynomial consists of a single segment of slope − 1 n {\displaystyle -{\frac {1}{n}}} connecting ( 0 , 1 ) {\displaystyle (0,1)} and ( n , 0 ) {\displaystyle (n,0)} , Eisenstein criterion follows. Indeed, by the main theorem, if α {\displaystyle \alpha } is a root of f {\displaystyle f} , v L ( α ) = − a / n . {\displaystyle v_{L}(\alpha )=-a/n.} If f {\displaystyle f} were not irreducible over K {\displaystyle K} , then the degree d {\displaystyle d} of α {\displaystyle \alpha } would be < n {\displaystyle <n} , and there would hold v L ( α ) ∈ 1 d Z {\displaystyle v_{L}(\alpha )\in {1 \over d}\mathbb {Z} } . But this is impossible since v L ( α ) = − a / n {\displaystyle v_{L}(\alpha )=-a/n} with a {\displaystyle a} coprime to n {\displaystyle n} . Another simple corollary is the following: Assume that ( K , v K ) {\displaystyle (K,v_{K})} is Henselian. If the Newton polygon of f {\displaystyle f} fulfills λ i = 1 {\displaystyle \lambda _{i}=1} for some i {\displaystyle i} , then f {\displaystyle f} has a root in K {\displaystyle K} . Proof: By the main theorem, f {\displaystyle f} must have a single root α {\displaystyle \alpha } whose valuation is v L ( α ) = − μ i . {\displaystyle v_{L}(\alpha )=-\mu _{i}.} In particular, α {\displaystyle \alpha } is separable over K {\displaystyle K} . If α {\displaystyle \alpha } does not belong to K {\displaystyle K} , α {\displaystyle \alpha } has a distinct Galois conjugate α ′ {\displaystyle \alpha '} over K {\displaystyle K} , with v L ( α ′ ) = v L ( α ) {\displaystyle v_{L}(\alpha ')=v_{L}(\alpha )} , and α ′ {\displaystyle \alpha '} is a root of f {\displaystyle f} , a contradiction. More generally, the following factorization theorem holds: Assume that ( K , v K ) {\displaystyle (K,v_{K})} is Henselian. Then f = A f 1 f 2 ⋯ f r , {\displaystyle f=A\,f_{1}\,f_{2}\cdots f_{r},} , where A ∈ K {\displaystyle A\in K} , f i ∈ K [ X ] {\displaystyle f_{i}\in K[X]} is monic for every i {\displaystyle i} , the roots of f i {\displaystyle f_{i}} are of valuation − μ i {\displaystyle -\mu _{i}} , and deg ( f i ) = λ i {\displaystyle \deg(f_{i})=\lambda _{i}} . Moreover, μ i = v K ( f i ( 0 ) ) / λ i {\displaystyle \mu _{i}=v_{K}(f_{i}(0))/\lambda _{i}} , and if v K ( f i ( 0 ) ) {\displaystyle v_{K}(f_{i}(0))} is coprime to λ i {\displaystyle \lambda _{i}} , f i {\displaystyle f_{i}} is irreducible over K {\displaystyle K} . Proof: For every i {\displaystyle i} , denote by f i {\displaystyle f_{i}} the product of the monomials ( X − α ) {\displaystyle (X-\alpha )} such that α {\displaystyle \alpha } is a root of f {\displaystyle f} and v L ( α ) = − μ i {\displaystyle v_{L}(\alpha )=-\mu _{i}} . We also denote f = A P 1 k 1 P 2 k 2 ⋯ P s k s {\displaystyle f=AP_{1}^{k_{1}}P_{2}^{k_{2}}\cdots P_{s}^{k_{s}}} the factorization of f {\displaystyle f} in K [ X ] {\displaystyle K[X]} into prime monic factors ( A ∈ K ) . {\displaystyle (A\in K).} Let α {\displaystyle \alpha } be a root of f i {\displaystyle f_{i}} . We can assume that P 1 {\displaystyle P_{1}} is the minimal polynomial of α {\displaystyle \alpha } over K {\displaystyle K} . If α ′ {\displaystyle \alpha '} is a root of P 1 {\displaystyle P_{1}} , there exists a K-automorphism σ {\displaystyle \sigma } of L {\displaystyle L} that sends α {\displaystyle \alpha } to α ′ {\displaystyle \alpha '} , and we have v L ( σ α ) = v L ( α ) {\displaystyle v_{L}(\sigma \alpha )=v_{L}(\alpha )} since K {\displaystyle K} is Henselian. Therefore α ′ {\displaystyle \alpha '} is also a root of f i {\displaystyle f_{i}} . Moreover, every root of P 1 {\displaystyle P_{1}} of multiplicity ν {\displaystyle \nu } is clearly a root of f i {\displaystyle f_{i}} of multiplicity k 1 ν {\displaystyle k_{1}\nu } , since repeated roots share obviously the same valuation. This shows that P 1 k 1 {\displaystyle P_{1}^{k_{1}}} divides f i . {\displaystyle f_{i}.} Let g i = f i / P 1 k 1 {\displaystyle g_{i}=f_{i}/P_{1}^{k_{1}}} . Choose a root β {\displaystyle \beta } of g i {\displaystyle g_{i}} . Notice that the roots of g i {\displaystyle g_{i}} are distinct from the roots of P 1 {\displaystyle P_{1}} . Repeat the previous argument with the minimal polynomial of β {\displaystyle \beta } over K {\displaystyle K} , assumed w.l.g. to be P 2 {\displaystyle P_{2}} , to show that P 2 k 2 {\displaystyle P_{2}^{k_{2}}} divides g i {\displaystyle g_{i}} . Continuing this process until all the roots of f i {\displaystyle f_{i}} are exhausted, one eventually arrives to f i = P 1 k 1 ⋯ P m k m {\displaystyle f_{i}=P_{1}^{k_{1}}\cdots P_{m}^{k_{m}}} , with m ≤ s {\displaystyle m\leq s} . This shows that f i ∈ K [ X ] {\displaystyle f_{i}\in K[X]} , f i {\displaystyle f_{i}} monic. But the f i {\displaystyle f_{i}} are coprime since their roots have distinct valuations. Hence clearly f = A f 1 ⋅ f 2 ⋯ f r {\displaystyle f=Af_{1}\cdot f_{2}\cdots f_{r}} , showing the main contention. The fact that λ i = deg ( f i ) {\displaystyle \lambda _{i}=\deg(f_{i})} follows from the main theorem, and so does the fact that μ i = v K ( f i ( 0 ) ) / λ i {\displaystyle \mu _{i}=v_{K}(f_{i}(0))/\lambda _{i}} , by remarking that the Newton polygon of f i {\displaystyle f_{i}} can have only one segment joining ( 0 , v K ( f i ( 0 ) ) {\displaystyle (0,v_{K}(f_{i}(0))} to ( λ i , 0 = v K ( 1 ) ) {\displaystyle (\lambda _{i},0=v_{K}(1))} . The condition for the irreducibility of f i {\displaystyle f_{i}} follows from the corollary above. (q.e.d.) The following is an immediate corollary of the factorization above, and constitutes a test for the reducibility of polynomials over Henselian fields: Assume that ( K , v K ) {\displaystyle (K,v_{K})} is Henselian. If the Newton polygon does not reduce to a single segment ( μ , λ ) , {\displaystyle (\mu ,\lambda ),} then f {\displaystyle f} is reducible over K {\displaystyle K} . Other applications of the Newton polygon comes from the fact that a Newton Polygon is sometimes a special case of a Newton polytope, and can be used to construct asymptotic solutions of two-variable polynomial equations like 3 x 2 y 3 − x y 2 + 2 x 2 y 2 − x 3 y = 0. {\displaystyle 3x^{2}y^{3}-xy^{2}+2x^{2}y^{2}-x^{3}y=0.} == Symmetric function explanation == In the context of a valuation, we are given certain information in the form of the valuations of elementary symmetric functions of the roots of a polynomial, and require information on the valuations of the actual roots, in an algebraic closure. This has aspects both of ramification theory and singularity theory. The valid inferences possible are to the valuations of power sums, by means of Newton's identities. == History == Newton polygons are named after Isaac Newton, who first described them and some of their uses in correspondence from the year 1676 addressed to Henry Oldenburg. == See also == F-crystal Eisenstein's criterion Newton–Okounkov body Newton polytope == References == Goss, David (1996), Basic structures of function field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 35, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-61480-4, ISBN 978-3-540-61087-8, MR 1423131 Gouvêa, Fernando: p-adic numbers: An introduction. Springer Verlag 1993. p. 199. == External links == Applet drawing a Newton Polygon
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Wikipedia:Newton polynomial#0
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In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using Newton's divided differences method. == Definition == Given a set of k + 1 data points ( x 0 , y 0 ) , … , ( x j , y j ) , … , ( x k , y k ) {\displaystyle (x_{0},y_{0}),\ldots ,(x_{j},y_{j}),\ldots ,(x_{k},y_{k})} where no two xj are the same, the Newton interpolation polynomial is a linear combination of Newton basis polynomials N ( x ) := ∑ j = 0 k a j n j ( x ) {\displaystyle N(x):=\sum _{j=0}^{k}a_{j}n_{j}(x)} with the Newton basis polynomials defined as n j ( x ) := ∏ i = 0 j − 1 ( x − x i ) {\displaystyle n_{j}(x):=\prod _{i=0}^{j-1}(x-x_{i})} for j > 0 and n 0 ( x ) ≡ 1 {\displaystyle n_{0}(x)\equiv 1} . The coefficients are defined as a j := [ y 0 , … , y j ] {\displaystyle a_{j}:=[y_{0},\ldots ,y_{j}]} where [ y 0 , … , y j ] {\displaystyle [y_{0},\ldots ,y_{j}]} are the divided differences defined as [ y k ] := y k , k ∈ { 0 , … , n } [ y k , … , y k + j ] := [ y k + 1 , … , y k + j ] − [ y k , … , y k + j − 1 ] x k + j − x k , k ∈ { 0 , … , n − j } , j ∈ { 1 , … , n } . {\displaystyle {\begin{aligned}{\mathopen {[}}y_{k}]&:=y_{k},&&k\in \{0,\ldots ,n\}\\{\mathopen {[}}y_{k},\ldots ,y_{k+j}]&:={\frac {[y_{k+1},\ldots ,y_{k+j}]-[y_{k},\ldots ,y_{k+j-1}]}{x_{k+j}-x_{k}}},&&k\in \{0,\ldots ,n-j\},\ j\in \{1,\ldots ,n\}.\end{aligned}}} Thus the Newton polynomial can be written as N ( x ) = [ y 0 ] + [ y 0 , y 1 ] ( x − x 0 ) + ⋯ + [ y 0 , … , y k ] ( x − x 0 ) ( x − x 1 ) ⋯ ( x − x k − 1 ) . {\displaystyle N(x)=[y_{0}]+[y_{0},y_{1}](x-x_{0})+\cdots +[y_{0},\ldots ,y_{k}](x-x_{0})(x-x_{1})\cdots (x-x_{k-1}).} === Newton forward divided difference formula === The Newton polynomial can be expressed in a simplified form when x 0 , x 1 , … , x k {\displaystyle x_{0},x_{1},\dots ,x_{k}} are arranged consecutively with equal spacing. If x 0 , x 1 , … , x k {\displaystyle x_{0},x_{1},\dots ,x_{k}} are consecutively arranged and equally spaced with x i = x 0 + i h {\displaystyle {x}_{i}={x}_{0}+ih} for i = 0, 1, ..., k and some variable x is expressed as x = x 0 + s h {\displaystyle {x}={x}_{0}+sh} , then the difference x − x i {\displaystyle x-x_{i}} can be written as ( s − i ) h {\displaystyle (s-i)h} . So the Newton polynomial becomes N ( x ) = [ y 0 ] + [ y 0 , y 1 ] s h + ⋯ + [ y 0 , … , y k ] s ( s − 1 ) ⋯ ( s − k + 1 ) h k = ∑ i = 0 k s ( s − 1 ) ⋯ ( s − i + 1 ) h i [ y 0 , … , y i ] = ∑ i = 0 k ( s i ) i ! h i [ y 0 , … , y i ] . {\displaystyle {\begin{aligned}N(x)&=[y_{0}]+[y_{0},y_{1}]sh+\cdots +[y_{0},\ldots ,y_{k}]s(s-1)\cdots (s-k+1){h}^{k}\\&=\sum _{i=0}^{k}s(s-1)\cdots (s-i+1){h}^{i}[y_{0},\ldots ,y_{i}]\\&=\sum _{i=0}^{k}{s \choose i}i!{h}^{i}[y_{0},\ldots ,y_{i}].\end{aligned}}} This is called the Newton forward divided difference formula. === Newton backward divided difference formula === If the nodes are reordered as , the Newton polynomial becomes N ( x ) = [ y k ] + [ y k , y k − 1 ] ( x − x k ) + ⋯ + [ y k , … , y 0 ] ( x − x k ) ( x − x k − 1 ) ⋯ ( x − x 1 ) . {\displaystyle N(x)=[y_{k}]+[{y}_{k},{y}_{k-1}](x-{x}_{k})+\cdots +[{y}_{k},\ldots ,{y}_{0}](x-{x}_{k})(x-{x}_{k-1})\cdots (x-{x}_{1}).} If x k , x k − 1 , … , x 0 {\displaystyle {x}_{k},\;{x}_{k-1},\;\dots ,\;{x}_{0}} are equally spaced with x i = x k − ( k − i ) h {\displaystyle {x}_{i}={x}_{k}-(k-i)h} for i = 0, 1, ..., k and x = x k + s h {\displaystyle {x}={x}_{k}+sh} , then, N ( x ) = [ y k ] + [ y k , y k − 1 ] s h + ⋯ + [ y k , … , y 0 ] s ( s + 1 ) ⋯ ( s + k − 1 ) h k = ∑ i = 0 k ( − 1 ) i ( − s i ) i ! h i [ y k , … , y k − i ] . {\displaystyle {\begin{aligned}N(x)&=[{y}_{k}]+[{y}_{k},{y}_{k-1}]sh+\cdots +[{y}_{k},\ldots ,{y}_{0}]s(s+1)\cdots (s+k-1){h}^{k}\\&=\sum _{i=0}^{k}{(-1)}^{i}{-s \choose i}i!{h}^{i}[{y}_{k},\ldots ,{y}_{k-i}].\end{aligned}}} This is called the Newton backward divided difference formula. == Significance == Newton's formula is of interest because it is the straightforward and natural differences-version of Taylor's polynomial. Taylor's polynomial tells where a function will go, based on its y value, and its derivatives (its rate of change, and the rate of change of its rate of change, etc.) at one particular x value. Newton's formula is Taylor's polynomial based on finite differences instead of instantaneous rates of change. === Polynomial interpolation === For a polynomial p n {\displaystyle p_{n}} of degree less than or equal to n, that interpolates f {\displaystyle f} at the nodes x i {\displaystyle x_{i}} where i = 0 , 1 , 2 , 3 , ⋯ , n {\displaystyle i=0,1,2,3,\cdots ,n} . Let p n + 1 {\displaystyle p_{n+1}} be the polynomial of degree less than or equal to n+1 that interpolates f {\displaystyle f} at the nodes x i {\displaystyle x_{i}} where i = 0 , 1 , 2 , 3 , ⋯ , n , n + 1 {\displaystyle i=0,1,2,3,\cdots ,n,n+1} . Then p n + 1 {\displaystyle p_{n+1}} is given by: p n + 1 ( x ) = p n ( x ) + a n + 1 w n ( x ) {\displaystyle p_{n+1}(x)=p_{n}(x)+a_{n+1}w_{n}(x)} where w n ( x ) := ∏ i = 0 n ( x − x i ) {\textstyle w_{n}(x):=\prod _{i=0}^{n}(x-x_{i})} and a n + 1 := f ( x n + 1 ) − p n ( x n + 1 ) w n ( x n + 1 ) {\textstyle a_{n+1}:={f(x_{n+1})-p_{n}(x_{n+1}) \over w_{n}(x_{n+1})}} . Proof: This can be shown for the case where i = 0 , 1 , 2 , 3 , ⋯ , n {\displaystyle i=0,1,2,3,\cdots ,n} : p n + 1 ( x i ) = p n ( x i ) + a n + 1 ∏ j = 0 n ( x i − x j ) = p n ( x i ) {\displaystyle p_{n+1}(x_{i})=p_{n}(x_{i})+a_{n+1}\prod _{j=0}^{n}(x_{i}-x_{j})=p_{n}(x_{i})} and when i = n + 1 {\displaystyle i=n+1} : p n + 1 ( x n + 1 ) = p n ( x n + 1 ) + f ( x n + 1 ) − p n ( x n + 1 ) w n ( x n + 1 ) w n ( x n + 1 ) = f ( x n + 1 ) {\displaystyle p_{n+1}(x_{n+1})=p_{n}(x_{n+1})+{f(x_{n+1})-p_{n}(x_{n+1}) \over w_{n}(x_{n+1})}w_{n}(x_{n+1})=f(x_{n+1})} By the uniqueness of interpolated polynomials of degree less than n + 1 {\displaystyle n+1} , p n + 1 ( x ) = p n ( x ) + a n + 1 w n ( x ) {\textstyle p_{n+1}(x)=p_{n}(x)+a_{n+1}w_{n}(x)} is the required polynomial interpolation. The function can thus be expressed as: p n ( x ) = a 0 + a 1 ( x − x 0 ) + a 2 ( x − x 0 ) ( x − x 1 ) + ⋯ + a n ( x − x 0 ) ⋯ ( x − x n − 1 ) {\textstyle p_{n}(x)=a_{0}+a_{1}(x-x_{0})+a_{2}(x-x_{0})(x-x_{1})+\cdots +a_{n}(x-x_{0})\cdots (x-x_{n-1})} where the factors a i {\displaystyle a_{i}} are divided differences. Thus, Newton polynomials are used to provide polynomial interpolation formula of n points. Taking y i = f ( x i ) {\displaystyle y_{i}=f(x_{i})} for some unknown function in Newton divided difference formulas, if the representation of x in the previous sections was instead taken to be x = x j + s h {\displaystyle x=x_{j}+sh} , in terms of forward differences, the Newton forward interpolation formula is expressed as: f ( x ) ≈ N ( x ) = N ( x j + s h ) = ∑ i = 0 k ( s i ) Δ ( i ) f ( x j ) {\displaystyle f(x)\approx N(x)=N(x_{j}+sh)=\sum _{i=0}^{k}{s \choose i}\Delta ^{(i)}f(x_{j})} whereas for the same in terms of backward differences, the Newton backward interpolation formula is expressed as: f ( x ) ≈ N ( x ) = N ( x j + s h ) = ∑ i = 0 k ( − 1 ) i ( − s i ) ∇ ( i ) f ( x j ) . {\displaystyle f(x)\approx N(x)=N(x_{j}+sh)=\sum _{i=0}^{k}{(-1)}^{i}{-s \choose i}\nabla ^{(i)}f(x_{j}).} This follows since relationship between divided differences and forward differences is given as: [ y j , y j + 1 , … , y j + n ] = 1 n ! h n Δ ( n ) y j , {\displaystyle [y_{j},y_{j+1},\ldots ,y_{j+n}]={\frac {1}{n!h^{n}}}\Delta ^{(n)}y_{j},} whereas for backward differences, it is given as: [ y j , y j − 1 , … , y j − n ] = 1 n ! h n ∇ ( n ) y j . {\displaystyle [{y}_{j},y_{j-1},\ldots ,{y}_{j-n}]={\frac {1}{n!h^{n}}}\nabla ^{(n)}y_{j}.} == Addition of new points == As with other difference formulas, the degree of a Newton interpolating polynomial can be increased by adding more terms and points without discarding existing ones. Newton's form has the simplicity that the new points are always added at one end: Newton's forward formula can add new points to the right, and Newton's backward formula can add new points to the left. The accuracy of polynomial interpolation depends on how close the interpolated point is to the middle of the x values of the set of points used. Obviously, as new points are added at one end, that middle becomes farther and farther from the first data point. Therefore, if it isn't known how many points will be needed for the desired accuracy, the middle of the x-values might be far from where the interpolation is done. Gauss, Stirling, and Bessel all developed formulae to remedy that problem. Gauss's formula alternately adds new points at the left and right ends, thereby keeping the set of points centered near the same place (near the evaluated point). When so doing, it uses terms from Newton's formula, with data points and x values renamed in keeping with one's choice of what data point is designated as the x0 data point. Stirling's formula remains centered about a particular data point, for use when the evaluated point is nearer to a data point than to a middle of two data points. Bessel's formula remains centered about a particular middle between two data points, for use when the evaluated point is nearer to a middle than to a data point. Bessel and Stirling achieve that by sometimes using the average of two differences, and sometimes using the average of two products of binomials in x, where Newton's or Gauss's would use just one difference or product. Stirling's uses an average difference in odd-degree terms (whose difference uses an even number of data points); Bessel's uses an average difference in even-degree terms (whose difference uses an odd number of data points). == Strengths and weaknesses of various formulae == For any given finite set of data points, there is only one polynomial of least possible degree that passes through all of them. Thus, it is appropriate to speak of the "Newton form", or Lagrange form, etc., of the interpolation polynomial. However, different methods of computing this polynomial can have differing computational efficiency. There are several similar methods, such as those of Gauss, Bessel and Stirling. They can be derived from Newton's by renaming the x-values of the data points, but in practice they are important. === Bessel vs. Stirling === The choice between Bessel and Stirling depends on whether the interpolated point is closer to a data point, or closer to a middle between two data points. A polynomial interpolation's error approaches zero, as the interpolation point approaches a data-point. Therefore, Stirling's formula brings its accuracy improvement where it is least needed and Bessel brings its accuracy improvement where it is most needed. So, Bessel's formula could be said to be the most consistently accurate difference formula, and, in general, the most consistently accurate of the familiar polynomial interpolation formulas. === Divided-Difference Methods vs. Lagrange === Lagrange is sometimes said to require less work, and is sometimes recommended for problems in which it is known, in advance, from previous experience, how many terms are needed for sufficient accuracy. The divided difference methods have the advantage that more data points can be added, for improved accuracy. The terms based on the previous data points can continue to be used. With the ordinary Lagrange formula, to do the problem with more data points would require re-doing the whole problem. There is a "barycentric" version of Lagrange that avoids the need to re-do the entire calculation when adding a new data point. But it requires that the values of each term be recorded. But the ability, of Gauss, Bessel and Stirling, to keep the data points centered close to the interpolated point gives them an advantage over Lagrange, when it isn't known, in advance, how many data points will be needed. Additionally, suppose that one wants to find out if, for some particular type of problem, linear interpolation is sufficiently accurate. That can be determined by evaluating the quadratic term of a divided difference formula. If the quadratic term is negligible—meaning that the linear term is sufficiently accurate without adding the quadratic term—then linear interpolation is sufficiently accurate. If the problem is sufficiently important, or if the quadratic term is nearly big enough to matter, then one might want to determine whether the sum of the quadratic and cubic terms is large enough to matter in the problem. Of course, only a divided-difference method can be used for such a determination. For that purpose, the divided-difference formula and/or its x0 point should be chosen so that the formula will use, for its linear term, the two data points between which the linear interpolation of interest would be done. The divided difference formulas are more versatile, useful in more kinds of problems. The Lagrange formula is at its best when all the interpolation will be done at one x value, with only the data points' y values varying from one problem to another, and when it is known, from past experience, how many terms are needed for sufficient accuracy. With the Newton form of the interpolating polynomial a compact and effective algorithm exists for combining the terms to find the coefficients of the polynomial. === Accuracy === When, with Stirling's or Bessel's, the last term used includes the average of two differences, then one more point is being used than Newton's or other polynomial interpolations would use for the same polynomial degree. So, in that instance, Stirling's or Bessel's is not putting an N−1 degree polynomial through N points, but is, instead, trading equivalence with Newton's for better centering and accuracy, giving those methods sometimes potentially greater accuracy, for a given polynomial degree, than other polynomial interpolations. == General case == For the special case of xi = i, there is a closely related set of polynomials, also called the Newton polynomials, that are simply the binomial coefficients for general argument. That is, one also has the Newton polynomials p n ( z ) {\displaystyle p_{n}(z)} given by p n ( z ) = ( z n ) = z ( z − 1 ) ⋯ ( z − n + 1 ) n ! {\displaystyle p_{n}(z)={z \choose n}={\frac {z(z-1)\cdots (z-n+1)}{n!}}} In this form, the Newton polynomials generate the Newton series. These are in turn a special case of the general difference polynomials which allow the representation of analytic functions through generalized difference equations. == Main idea == Solving an interpolation problem leads to a problem in linear algebra where we have to solve a system of linear equations. Using a standard monomial basis for our interpolation polynomial we get the very complicated Vandermonde matrix. By choosing another basis, the Newton basis, we get a system of linear equations with a much simpler lower triangular matrix which can be solved faster. For k + 1 data points we construct the Newton basis as n 0 ( x ) := 1 , n j ( x ) := ∏ i = 0 j − 1 ( x − x i ) j = 1 , … , k . {\displaystyle n_{0}(x):=1,\qquad n_{j}(x):=\prod _{i=0}^{j-1}(x-x_{i})\qquad j=1,\ldots ,k.} Using these polynomials as a basis for Π k {\displaystyle \Pi _{k}} we have to solve [ 1 … 0 1 x 1 − x 0 1 x 2 − x 0 ( x 2 − x 0 ) ( x 2 − x 1 ) ⋮ ⋮ ⋮ ⋱ 1 x k − x 0 … … ∏ j = 0 k − 1 ( x k − x j ) ] [ a 0 ⋮ a k ] = [ y 0 ⋮ y k ] {\displaystyle {\begin{bmatrix}1&&\ldots &&0\\1&x_{1}-x_{0}&&&\\1&x_{2}-x_{0}&(x_{2}-x_{0})(x_{2}-x_{1})&&\vdots \\\vdots &\vdots &&\ddots &\\1&x_{k}-x_{0}&\ldots &\ldots &\prod _{j=0}^{k-1}(x_{k}-x_{j})\end{bmatrix}}{\begin{bmatrix}a_{0}\\\\\vdots \\\\a_{k}\end{bmatrix}}={\begin{bmatrix}y_{0}\\\\\vdots \\\\y_{k}\end{bmatrix}}} to solve the polynomial interpolation problem. This system of equations can be solved iteratively by solving ∑ i = 0 j a i n i ( x j ) = y j j = 0 , … , k . {\displaystyle \sum _{i=0}^{j}a_{i}n_{i}(x_{j})=y_{j}\qquad j=0,\dots ,k.} == Derivation == While the interpolation formula can be found by solving a linear system of equations, there is a loss of intuition in what the formula is showing and why Newton's interpolation formula works is not readily apparent. To begin, we will need to establish two facts first: Fact 1. Reversing the terms of a divided difference leaves it unchanged: [ y 0 , … , y n ] = [ y n , … , y 0 ] . {\displaystyle [y_{0},\ldots ,y_{n}]=[y_{n},\ldots ,y_{0}].} The proof of this is an easy induction: for n = 1 {\displaystyle n=1} we compute [ y 0 , y 1 ] = [ y 1 ] − [ y 0 ] x 1 − x 0 = [ y 0 ] − [ y 1 ] x 0 − x 1 = [ y 1 , y 0 ] . {\displaystyle [y_{0},y_{1}]={\frac {[y_{1}]-[y_{0}]}{x_{1}-x_{0}}}={\frac {[y_{0}]-[y_{1}]}{x_{0}-x_{1}}}=[y_{1},y_{0}].} Induction step: Suppose the result holds for any divided difference involving at most n + 1 {\displaystyle n+1} terms. Then using the induction hypothesis in the following 2nd equality we see that for a divided difference involving n + 2 {\displaystyle n+2} terms we have [ y 0 , … , y n + 1 ] = [ y 1 , … , y n + 1 ] − [ y 0 , … , y n ] x n + 1 − x 0 = [ y n , … , y 0 ] − [ y n + 1 , … , y 1 ] x 0 − x n + 1 = [ y n + 1 , … , y 0 ] . {\displaystyle [y_{0},\ldots ,y_{n+1}]={\frac {[y_{1},\ldots ,y_{n+1}]-[y_{0},\ldots ,y_{n}]}{x_{n+1}-x_{0}}}={\frac {[y_{n},\ldots ,y_{0}]-[y_{n+1},\ldots ,y_{1}]}{x_{0}-x_{n+1}}}=[y_{n+1},\ldots ,y_{0}].} We formulate next Fact 2 which for purposes of induction and clarity we also call Statement n {\displaystyle n} ( Stm n {\displaystyle {\text{Stm}}_{n}} ) : Fact 2. ( Stm n {\displaystyle {\text{Stm}}_{n}} ) : If ( x 0 , y 0 ) , … , ( x n − 1 , y n − 1 ) {\displaystyle (x_{0},y_{0}),\ldots ,(x_{n-1},y_{n-1})} are any n {\displaystyle n} points with distinct x {\displaystyle x} -coordinates and P = P ( x ) {\displaystyle P=P(x)} is the unique polynomial of degree (at most) n − 1 {\displaystyle n-1} whose graph passes through these n {\displaystyle n} points then there holds the relation [ y 0 , … , y n ] ( x n − x 0 ) ⋅ … ⋅ ( x n − x n − 1 ) = y n − P ( x n ) {\displaystyle [y_{0},\ldots ,y_{n}](x_{n}-x_{0})\cdot \ldots \cdot (x_{n}-x_{n-1})=y_{n}-P(x_{n})} Proof. (It will be helpful for fluent reading of the proof to have the precise statement and its subtlety in mind: P {\displaystyle P} is defined by passing through ( x 0 , y 0 ) , . . . , ( x n − 1 , y n − 1 ) {\displaystyle (x_{0},y_{0}),...,(x_{n-1},y_{n-1})} but the formula also speaks at both sides of an additional arbitrary point ( x n , y n ) {\displaystyle (x_{n},y_{n})} with x {\displaystyle x} -coordinate distinct from the other x i {\displaystyle x_{i}} .) We again prove these statements by induction. To show Stm 1 , {\displaystyle {\text{Stm}}_{1},} let ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} be any one point and let P ( x ) {\displaystyle P(x)} be the unique polynomial of degree 0 passing through ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} . Then evidently P ( x ) = y 0 {\displaystyle P(x)=y_{0}} and we can write [ y 0 , y 1 ] ( x 1 − x 0 ) = y 1 − y 0 x 1 − x 0 ( x 1 − x 0 ) = y 1 − y 0 = y 1 − P ( x 1 ) {\displaystyle [y_{0},y_{1}](x_{1}-x_{0})={\frac {y_{1}-y_{0}}{x_{1}-x_{0}}}(x_{1}-x_{0})=y_{1}-y_{0}=y_{1}-P(x_{1})} as wanted. Proof of Stm n + 1 , {\displaystyle {\text{Stm}}_{n+1},} assuming Stm n {\displaystyle {\text{Stm}}_{n}} already established: Let P ( x ) {\displaystyle P(x)} be the polynomial of degree (at most) n {\displaystyle n} passing through ( x 0 , y 0 ) , … , ( x n , y n ) . {\displaystyle (x_{0},y_{0}),\ldots ,(x_{n},y_{n}).} With Q ( x ) {\displaystyle Q(x)} being the unique polynomial of degree (at most) n − 1 {\displaystyle n-1} passing through the points ( x 1 , y 1 ) , … , ( x n , y n ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})} , we can write the following chain of equalities, where we use in the penultimate equality that Stm n {\displaystyle _{n}} applies to Q {\displaystyle Q} : [ y 0 , … , y n + 1 ] ( x n + 1 − x 0 ) ⋅ … ⋅ ( x n + 1 − x n ) = [ y 1 , … , y n + 1 ] − [ y 0 , … , y n ] x n + 1 − x 0 ( x n + 1 − x 0 ) ⋅ … ⋅ ( x n + 1 − x n ) = ( [ y 1 , … , y n + 1 ] − [ y 0 , … , y n ] ) ( x n + 1 − x 1 ) ⋅ … ⋅ ( x n + 1 − x n ) = [ y 1 , … , y n + 1 ] ( x n + 1 − x 1 ) ⋅ … ⋅ ( x n + 1 − x n ) − [ y 0 , … , y n ] ( x n + 1 − x 1 ) ⋅ … ⋅ ( x n + 1 − x n ) = ( y n + 1 − Q ( x n + 1 ) ) − [ y 0 , … , y n ] ( x n + 1 − x 1 ) ⋅ … ⋅ ( x n + 1 − x n ) = y n + 1 − ( Q ( x n + 1 ) + [ y 0 , … , y n ] ( x n + 1 − x 1 ) ⋅ … ⋅ ( x n + 1 − x n ) ) . {\displaystyle {\begin{aligned}&[y_{0},\ldots ,y_{n+1}](x_{n+1}-x_{0})\cdot \ldots \cdot (x_{n+1}-x_{n})\\&={\frac {[y_{1},\ldots ,y_{n+1}]-[y_{0},\ldots ,y_{n}]}{x_{n+1}-x_{0}}}(x_{n+1}-x_{0})\cdot \ldots \cdot (x_{n+1}-x_{n})\\&=\left([y_{1},\ldots ,y_{n+1}]-[y_{0},\ldots ,y_{n}]\right)(x_{n+1}-x_{1})\cdot \ldots \cdot (x_{n+1}-x_{n})\\&=[y_{1},\ldots ,y_{n+1}](x_{n+1}-x_{1})\cdot \ldots \cdot (x_{n+1}-x_{n})-[y_{0},\ldots ,y_{n}](x_{n+1}-x_{1})\cdot \ldots \cdot (x_{n+1}-x_{n})\\&=(y_{n+1}-Q(x_{n+1}))-[y_{0},\ldots ,y_{n}](x_{n+1}-x_{1})\cdot \ldots \cdot (x_{n+1}-x_{n})\\&=y_{n+1}-(Q(x_{n+1})+[y_{0},\ldots ,y_{n}](x_{n+1}-x_{1})\cdot \ldots \cdot (x_{n+1}-x_{n})).\end{aligned}}} The induction hypothesis for Q {\displaystyle Q} also applies to the second equality in the following computation, where ( x 0 , y 0 ) {\displaystyle (x_{0},y_{0})} is added to the points defining Q {\displaystyle Q} : Q ( x 0 ) + [ y 0 , … , y n ] ( x 0 − x 1 ) ⋅ … ⋅ ( x 0 − x n ) = Q ( x 0 ) + [ y n , … , y 0 ] ( x 0 − x n ) ⋅ … ⋅ ( x 0 − x 1 ) = Q ( x 0 ) + y 0 − Q ( x 0 ) = y 0 = P ( x 0 ) . {\displaystyle {\begin{aligned}&Q(x_{0})+[y_{0},\ldots ,y_{n}](x_{0}-x_{1})\cdot \ldots \cdot (x_{0}-x_{n})\\&=Q(x_{0})+[y_{n},\ldots ,y_{0}](x_{0}-x_{n})\cdot \ldots \cdot (x_{0}-x_{1})\\&=Q(x_{0})+y_{0}-Q(x_{0})\\&=y_{0}\\&=P(x_{0}).\\\end{aligned}}} Now look at Q ( x ) + [ y 0 , … , y n ] ( x − x 1 ) ⋅ … ⋅ ( x − x n ) . {\displaystyle Q(x)+[y_{0},\ldots ,y_{n}](x-x_{1})\cdot \ldots \cdot (x-x_{n}).} By the definition of Q {\displaystyle Q} this polynomial passes through ( x 1 , y 1 ) , . . . , ( x n , y n ) {\displaystyle (x_{1},y_{1}),...,(x_{n},y_{n})} and, as we have just shown, it also passes through ( x 0 , y 0 ) . {\displaystyle (x_{0},y_{0}).} Thus it is the unique polynomial of degree ≤ n {\displaystyle \leq n} which passes through these points. Therefore this polynomial is P ( x ) ; {\displaystyle P(x);} i.e.: P ( x ) = Q ( x ) + [ y 0 , … , y n ] ( x − x 1 ) ⋅ … ⋅ ( x − x n ) . {\displaystyle P(x)=Q(x)+[y_{0},\ldots ,y_{n}](x-x_{1})\cdot \ldots \cdot (x-x_{n}).} Thus we can write the last line in the first chain of equalities as ` y n + 1 − P ( x n + 1 ) {\displaystyle y_{n+1}-P(x_{n+1})} ' and have thus established that [ y 0 , … , y n + 1 ] ( x n + 1 − x 0 ) ⋅ … ⋅ ( x n + 1 − x n ) = y n + 1 − P ( x n + 1 ) . {\displaystyle [y_{0},\ldots ,y_{n+1}](x_{n+1}-x_{0})\cdot \ldots \cdot (x_{n+1}-x_{n})=y_{n+1}-P(x_{n+1}).} So we established Stm n + 1 {\displaystyle {\text{Stm}}_{n+1}} , and hence completed the proof of Fact 2. Now look at Fact 2: It can be formulated this way: If P {\displaystyle P} is the unique polynomial of degree at most n − 1 {\displaystyle n-1} whose graph passes through the points ( x 0 , y 0 ) , . . . , ( x n − 1 , y n − 1 ) , {\displaystyle (x_{0},y_{0}),...,(x_{n-1},y_{n-1}),} then P ( x ) + [ y 0 , … , y n ] ( x − x 0 ) ⋅ … ⋅ ( x − x n − 1 ) {\displaystyle P(x)+[y_{0},\ldots ,y_{n}](x-x_{0})\cdot \ldots \cdot (x-x_{n-1})} is the unique polynomial of degree at most n {\displaystyle n} passing through points ( x 0 , y 0 ) , . . . , ( x n − 1 , y n − 1 ) , ( x n , y n ) . {\displaystyle (x_{0},y_{0}),...,(x_{n-1},y_{n-1}),(x_{n},y_{n}).} So we see Newton interpolation permits indeed to add new interpolation points without destroying what has already been computed. == Taylor polynomial == The limit of the Newton polynomial if all nodes coincide is a Taylor polynomial, because the divided differences become derivatives. lim ( x 0 , … , x n ) → ( z , … , z ) f [ x 0 ] + f [ x 0 , x 1 ] ⋅ ( ξ − x 0 ) + ⋯ + f [ x 0 , … , x n ] ⋅ ( ξ − x 0 ) ⋅ ⋯ ⋅ ( ξ − x n − 1 ) = f ( z ) + f ′ ( z ) ⋅ ( ξ − z ) + ⋯ + f ( n ) ( z ) n ! ⋅ ( ξ − z ) n {\displaystyle {\begin{aligned}&\lim _{(x_{0},\dots ,x_{n})\to (z,\dots ,z)}f[x_{0}]+f[x_{0},x_{1}]\cdot (\xi -x_{0})+\dots +f[x_{0},\dots ,x_{n}]\cdot (\xi -x_{0})\cdot \dots \cdot (\xi -x_{n-1})\\&=f(z)+f'(z)\cdot (\xi -z)+\dots +{\frac {f^{(n)}(z)}{n!}}\cdot (\xi -z)^{n}\end{aligned}}} == Application == As can be seen from the definition of the divided differences new data points can be added to the data set to create a new interpolation polynomial without recalculating the old coefficients. And when a data point changes we usually do not have to recalculate all coefficients. Furthermore, if the xi are distributed equidistantly the calculation of the divided differences becomes significantly easier. Therefore, the divided-difference formulas are usually preferred over the Lagrange form for practical purposes. === Examples === The divided differences can be written in the form of a table. For example, for a function f is to be interpolated on points x 0 , … , x n {\displaystyle x_{0},\ldots ,x_{n}} . Write x 0 f ( x 0 ) f ( x 1 ) − f ( x 0 ) x 1 − x 0 x 1 f ( x 1 ) f ( x 2 ) − f ( x 1 ) x 2 − x 1 − f ( x 1 ) − f ( x 0 ) x 1 − x 0 x 2 − x 0 f ( x 2 ) − f ( x 1 ) x 2 − x 1 x 2 f ( x 2 ) ⋮ ⋮ ⋮ ⋮ ⋮ x n f ( x n ) {\displaystyle {\begin{matrix}x_{0}&f(x_{0})&&\\&&{f(x_{1})-f(x_{0}) \over x_{1}-x_{0}}&\\x_{1}&f(x_{1})&&{{f(x_{2})-f(x_{1}) \over x_{2}-x_{1}}-{f(x_{1})-f(x_{0}) \over x_{1}-x_{0}} \over x_{2}-x_{0}}\\&&{f(x_{2})-f(x_{1}) \over x_{2}-x_{1}}&\\x_{2}&f(x_{2})&&\vdots \\&&\vdots &\\\vdots &&&\vdots \\&&\vdots &\\x_{n}&f(x_{n})&&\\\end{matrix}}} Then the interpolating polynomial is formed as above using the topmost entries in each column as coefficients. For example, suppose we are to construct the interpolating polynomial to f(x) = tan(x) using divided differences, at the points Using six digits of accuracy, we construct the table − 3 2 − 14.1014 17.5597 − 3 4 − 0.931596 − 10.8784 1.24213 4.83484 0 0 0 0 1.24213 4.83484 3 4 0.931596 10.8784 17.5597 3 2 14.1014 {\displaystyle {\begin{matrix}-{\tfrac {3}{2}}&-14.1014&&&&\\&&17.5597&&&\\-{\tfrac {3}{4}}&-0.931596&&-10.8784&&\\&&1.24213&&4.83484&\\0&0&&0&&0\\&&1.24213&&4.83484&\\{\tfrac {3}{4}}&0.931596&&10.8784&&\\&&17.5597&&&\\{\tfrac {3}{2}}&14.1014&&&&\\\end{matrix}}} Thus, the interpolating polynomial is − 14.1014 + 17.5597 ( x + 3 2 ) − 10.8784 ( x + 3 2 ) ( x + 3 4 ) + 4.83484 ( x + 3 2 ) ( x + 3 4 ) ( x ) + 0 ( x + 3 2 ) ( x + 3 4 ) ( x ) ( x − 3 4 ) = − 0.00005 − 1.4775 x − 0.00001 x 2 + 4.83484 x 3 {\displaystyle {\begin{aligned}&-14.1014+17.5597(x+{\tfrac {3}{2}})-10.8784(x+{\tfrac {3}{2}})(x+{\tfrac {3}{4}})+4.83484(x+{\tfrac {3}{2}})(x+{\tfrac {3}{4}})(x)+0(x+{\tfrac {3}{2}})(x+{\tfrac {3}{4}})(x)(x-{\tfrac {3}{4}})\\={}&-0.00005-1.4775x-0.00001x^{2}+4.83484x^{3}\end{aligned}}} Given more digits of accuracy in the table, the first and third coefficients will be found to be zero. Another example: The sequence f 0 {\displaystyle f_{0}} such that f 0 ( 1 ) = 6 , f 0 ( 2 ) = 9 , f 0 ( 3 ) = 2 {\displaystyle f_{0}(1)=6,f_{0}(2)=9,f_{0}(3)=2} and f 0 ( 4 ) = 5 {\displaystyle f_{0}(4)=5} , i.e., they are 6 , 9 , 2 , 5 {\displaystyle 6,9,2,5} from x 0 = 1 {\displaystyle x_{0}=1} to x 3 = 4 {\displaystyle x_{3}=4} . You obtain the slope of order 1 {\displaystyle 1} in the following way: f 1 ( x 0 , x 1 ) = f 0 ( x 1 ) − f 0 ( x 0 ) x 1 − x 0 = 9 − 6 2 − 1 = 3 {\displaystyle f_{1}(x_{0},x_{1})={\frac {f_{0}(x_{1})-f_{0}(x_{0})}{x_{1}-x_{0}}}={\frac {9-6}{2-1}}=3} f 1 ( x 1 , x 2 ) = f 0 ( x 2 ) − f 0 ( x 1 ) x 2 − x 1 = 2 − 9 3 − 2 = − 7 {\displaystyle f_{1}(x_{1},x_{2})={\frac {f_{0}(x_{2})-f_{0}(x_{1})}{x_{2}-x_{1}}}={\frac {2-9}{3-2}}=-7} f 1 ( x 2 , x 3 ) = f 0 ( x 3 ) − f 0 ( x 2 ) x 3 − x 2 = 5 − 2 4 − 3 = 3 {\displaystyle f_{1}(x_{2},x_{3})={\frac {f_{0}(x_{3})-f_{0}(x_{2})}{x_{3}-x_{2}}}={\frac {5-2}{4-3}}=3} As we have the slopes of order 1 {\displaystyle 1} , it is possible to obtain the next order: f 2 ( x 0 , x 1 , x 2 ) = f 1 ( x 1 , x 2 ) − f 1 ( x 0 , x 1 ) x 2 − x 0 = − 7 − 3 3 − 1 = − 5 {\displaystyle f_{2}(x_{0},x_{1},x_{2})={\frac {f_{1}(x_{1},x_{2})-f_{1}(x_{0},x_{1})}{x_{2}-x_{0}}}={\frac {-7-3}{3-1}}=-5} f 2 ( x 1 , x 2 , x 3 ) = f 1 ( x 2 , x 3 ) − f 1 ( x 1 , x 2 ) x 3 − x 1 = 3 − ( − 7 ) 4 − 2 = 5 {\displaystyle f_{2}(x_{1},x_{2},x_{3})={\frac {f_{1}(x_{2},x_{3})-f_{1}(x_{1},x_{2})}{x_{3}-x_{1}}}={\frac {3-(-7)}{4-2}}=5} Finally, we define the slope of order 3 {\displaystyle 3} : f 3 ( x 0 , x 1 , x 2 , x 3 ) = f 2 ( x 1 , x 2 , x 3 ) − f 2 ( x 0 , x 1 , x 2 ) x 3 − x 0 = 5 − ( − 5 ) 4 − 1 = 10 3 {\displaystyle f_{3}(x_{0},x_{1},x_{2},x_{3})={\frac {f_{2}(x_{1},x_{2},x_{3})-f_{2}(x_{0},x_{1},x_{2})}{x_{3}-x_{0}}}={\frac {5-(-5)}{4-1}}={\frac {10}{3}}} Once we have the slope, we can define the consequent polynomials: p 0 ( x ) = 6 {\displaystyle p_{0}(x)=6} . p 1 ( x ) = 6 + 3 ( x − 1 ) {\displaystyle p_{1}(x)=6+3(x-1)} p 2 ( x ) = 6 + 3 ( x − 1 ) − 5 ( x − 1 ) ( x − 2 ) {\displaystyle p_{2}(x)=6+3(x-1)-5(x-1)(x-2)} . p 3 ( x ) = 6 + 3 ( x − 1 ) − 5 ( x − 1 ) ( x − 2 ) + 10 3 ( x − 1 ) ( x − 2 ) ( x − 3 ) {\displaystyle p_{3}(x)=6+3(x-1)-5(x-1)(x-2)+{\frac {10}{3}}(x-1)(x-2)(x-3)} == See also == De numeris triangularibus et inde de progressionibus arithmeticis: Magisteria magna, a work by Thomas Harriot describing similar methods for interpolation, written 50 years earlier than Newton's work but not published until 2009 Newton series Neville's schema Polynomial interpolation Lagrange form of the interpolation polynomial Bernstein form of the interpolation polynomial Hermite interpolation Carlson's theorem Table of Newtonian series == References == == External links == Module for the Newton Polynomial by John H. Mathews
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Wikipedia:Newton's identities#0
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In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of all roots of P (counted with their multiplicity) in terms of the coefficients of P, without actually finding those roots. These identities were found by Isaac Newton around 1666, apparently in ignorance of earlier work (1629) by Albert Girard. They have applications in many areas of mathematics, including Galois theory, invariant theory, group theory, combinatorics, as well as further applications outside mathematics, including general relativity. == Mathematical statement == === Formulation in terms of symmetric polynomials === Let x1, ..., xn be variables, denote for k ≥ 1 by pk(x1, ..., xn) the k-th power sum: p k ( x 1 , … , x n ) = ∑ i = 1 n x i k = x 1 k + ⋯ + x n k , {\displaystyle p_{k}(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}x_{i}^{k}=x_{1}^{k}+\cdots +x_{n}^{k},} and for k ≥ 0 denote by ek(x1, ..., xn) the elementary symmetric polynomial (that is, the sum of all distinct products of k distinct variables), so e 0 ( x 1 , … , x n ) = 1 , e 1 ( x 1 , … , x n ) = x 1 + x 2 + ⋯ + x n , e 2 ( x 1 , … , x n ) = ∑ 1 ≤ i < j ≤ n x i x j , ⋮ e n ( x 1 , … , x n ) = x 1 x 2 ⋯ x n , e k ( x 1 , … , x n ) = 0 , for k > n . {\displaystyle {\begin{aligned}e_{0}(x_{1},\ldots ,x_{n})&=1,\\e_{1}(x_{1},\ldots ,x_{n})&=x_{1}+x_{2}+\cdots +x_{n},\\e_{2}(x_{1},\ldots ,x_{n})&=\sum _{1\leq i<j\leq n}x_{i}x_{j},\\&\;\;\vdots \\e_{n}(x_{1},\ldots ,x_{n})&=x_{1}x_{2}\cdots x_{n},\\e_{k}(x_{1},\ldots ,x_{n})&=0,\quad {\text{for}}\ k>n.\\\end{aligned}}} Then Newton's identities can be stated as k e k ( x 1 , … , x n ) = ∑ i = 1 k ( − 1 ) i − 1 e k − i ( x 1 , … , x n ) p i ( x 1 , … , x n ) , {\displaystyle ke_{k}(x_{1},\ldots ,x_{n})=\sum _{i=1}^{k}(-1)^{i-1}e_{k-i}(x_{1},\ldots ,x_{n})p_{i}(x_{1},\ldots ,x_{n}),} valid for all n ≥ k ≥ 1. Also, one has 0 = ∑ i = k − n k ( − 1 ) i − 1 e k − i ( x 1 , … , x n ) p i ( x 1 , … , x n ) , {\displaystyle 0=\sum _{i=k-n}^{k}(-1)^{i-1}e_{k-i}(x_{1},\ldots ,x_{n})p_{i}(x_{1},\ldots ,x_{n}),} for all k > n ≥ 1. Concretely, one gets for the first few values of k: e 1 ( x 1 , … , x n ) = p 1 ( x 1 , … , x n ) , 2 e 2 ( x 1 , … , x n ) = e 1 ( x 1 , … , x n ) p 1 ( x 1 , … , x n ) − p 2 ( x 1 , … , x n ) , 3 e 3 ( x 1 , … , x n ) = e 2 ( x 1 , … , x n ) p 1 ( x 1 , … , x n ) − e 1 ( x 1 , … , x n ) p 2 ( x 1 , … , x n ) + p 3 ( x 1 , … , x n ) . {\displaystyle {\begin{aligned}e_{1}(x_{1},\ldots ,x_{n})&=p_{1}(x_{1},\ldots ,x_{n}),\\2e_{2}(x_{1},\ldots ,x_{n})&=e_{1}(x_{1},\ldots ,x_{n})p_{1}(x_{1},\ldots ,x_{n})-p_{2}(x_{1},\ldots ,x_{n}),\\3e_{3}(x_{1},\ldots ,x_{n})&=e_{2}(x_{1},\ldots ,x_{n})p_{1}(x_{1},\ldots ,x_{n})-e_{1}(x_{1},\ldots ,x_{n})p_{2}(x_{1},\ldots ,x_{n})+p_{3}(x_{1},\ldots ,x_{n}).\end{aligned}}} The form and validity of these equations do not depend on the number n of variables (although the point where the left-hand side becomes 0 does, namely after the n-th identity), which makes it possible to state them as identities in the ring of symmetric functions. In that ring one has e 1 = p 1 , 2 e 2 = e 1 p 1 − p 2 = p 1 2 − p 2 , 3 e 3 = e 2 p 1 − e 1 p 2 + p 3 = 1 2 p 1 3 − 3 2 p 1 p 2 + p 3 , 4 e 4 = e 3 p 1 − e 2 p 2 + e 1 p 3 − p 4 = 1 6 p 1 4 − p 1 2 p 2 + 4 3 p 1 p 3 + 1 2 p 2 2 − p 4 , {\displaystyle {\begin{aligned}e_{1}&=p_{1},\\2e_{2}&=e_{1}p_{1}-p_{2}=p_{1}^{2}-p_{2},\\3e_{3}&=e_{2}p_{1}-e_{1}p_{2}+p_{3}={\tfrac {1}{2}}p_{1}^{3}-{\tfrac {3}{2}}p_{1}p_{2}+p_{3},\\4e_{4}&=e_{3}p_{1}-e_{2}p_{2}+e_{1}p_{3}-p_{4}={\tfrac {1}{6}}p_{1}^{4}-p_{1}^{2}p_{2}+{\tfrac {4}{3}}p_{1}p_{3}+{\tfrac {1}{2}}p_{2}^{2}-p_{4},\\\end{aligned}}} and so on; here the left-hand sides never become zero. These equations allow to recursively express the ei in terms of the pk; to be able to do the inverse, one may rewrite them as p 1 = e 1 , p 2 = e 1 p 1 − 2 e 2 = e 1 2 − 2 e 2 , p 3 = e 1 p 2 − e 2 p 1 + 3 e 3 = e 1 3 − 3 e 1 e 2 + 3 e 3 , p 4 = e 1 p 3 − e 2 p 2 + e 3 p 1 − 4 e 4 = e 1 4 − 4 e 1 2 e 2 + 4 e 1 e 3 + 2 e 2 2 − 4 e 4 , ⋮ {\displaystyle {\begin{aligned}p_{1}&=e_{1},\\p_{2}&=e_{1}p_{1}-2e_{2}=e_{1}^{2}-2e_{2},\\p_{3}&=e_{1}p_{2}-e_{2}p_{1}+3e_{3}=e_{1}^{3}-3e_{1}e_{2}+3e_{3},\\p_{4}&=e_{1}p_{3}-e_{2}p_{2}+e_{3}p_{1}-4e_{4}=e_{1}^{4}-4e_{1}^{2}e_{2}+4e_{1}e_{3}+2e_{2}^{2}-4e_{4},\\&{}\ \ \vdots \end{aligned}}} In general, we have p k ( x 1 , … , x n ) = ( − 1 ) k − 1 k e k ( x 1 , … , x n ) + ∑ i = 1 k − 1 ( − 1 ) k − 1 + i e k − i ( x 1 , … , x n ) p i ( x 1 , … , x n ) , {\displaystyle p_{k}(x_{1},\ldots ,x_{n})=(-1)^{k-1}ke_{k}(x_{1},\ldots ,x_{n})+\sum _{i=1}^{k-1}(-1)^{k-1+i}e_{k-i}(x_{1},\ldots ,x_{n})p_{i}(x_{1},\ldots ,x_{n}),} valid for all n ≥k ≥ 1. Also, one has p k ( x 1 , … , x n ) = ∑ i = k − n k − 1 ( − 1 ) k − 1 + i e k − i ( x 1 , … , x n ) p i ( x 1 , … , x n ) , {\displaystyle p_{k}(x_{1},\ldots ,x_{n})=\sum _{i=k-n}^{k-1}(-1)^{k-1+i}e_{k-i}(x_{1},\ldots ,x_{n})p_{i}(x_{1},\ldots ,x_{n}),} for all k > n ≥ 1. === Application to the roots of a polynomial === The polynomial with roots xi may be expanded as ∏ i = 1 n ( x − x i ) = ∑ k = 0 n ( − 1 ) k e k x n − k , {\displaystyle \prod _{i=1}^{n}(x-x_{i})=\sum _{k=0}^{n}(-1)^{k}e_{k}x^{n-k},} where the coefficients e k ( x 1 , … , x n ) {\displaystyle e_{k}(x_{1},\ldots ,x_{n})} are the symmetric polynomials defined above. Given the power sums of the roots p k ( x 1 , … , x n ) = ∑ i = 1 n x i k , {\displaystyle p_{k}(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}x_{i}^{k},} the coefficients of the polynomial with roots x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} may be expressed recursively in terms of the power sums as e 0 = 1 , − e 1 = − p 1 , e 2 = 1 2 ( e 1 p 1 − p 2 ) , − e 3 = − 1 3 ( e 2 p 1 − e 1 p 2 + p 3 ) , e 4 = 1 4 ( e 3 p 1 − e 2 p 2 + e 1 p 3 − p 4 ) , ⋮ {\displaystyle {\begin{aligned}e_{0}&=1,\\[4pt]-e_{1}&=-p_{1},\\[4pt]e_{2}&={\frac {1}{2}}(e_{1}p_{1}-p_{2}),\\[4pt]-e_{3}&=-{\frac {1}{3}}(e_{2}p_{1}-e_{1}p_{2}+p_{3}),\\[4pt]e_{4}&={\frac {1}{4}}(e_{3}p_{1}-e_{2}p_{2}+e_{1}p_{3}-p_{4}),\\&{}\ \ \vdots \end{aligned}}} Formulating polynomials in this way is useful in using the method of Delves and Lyness to find the zeros of an analytic function. === Application to the characteristic polynomial of a matrix === When the polynomial above is the characteristic polynomial of a matrix A {\displaystyle \mathbf {A} } (in particular when A {\displaystyle \mathbf {A} } is the companion matrix of the polynomial), the roots x i {\displaystyle x_{i}} are the eigenvalues of the matrix, counted with their algebraic multiplicity. For any positive integer k {\displaystyle k} , the matrix A k {\displaystyle \mathbf {A} ^{k}} has as eigenvalues the powers x i k {\displaystyle x_{i}^{k}} , and each eigenvalue x i {\displaystyle x_{i}} of A {\displaystyle \mathbf {A} } contributes its multiplicity to that of the eigenvalue x i k {\displaystyle x_{i}^{k}} of A k {\displaystyle \mathbf {A} ^{k}} . Then the coefficients of the characteristic polynomial of A k {\displaystyle \mathbf {A} ^{k}} are given by the elementary symmetric polynomials in those powers x i k {\displaystyle x_{i}^{k}} . In particular, the sum of the x i k {\displaystyle x_{i}^{k}} , which is the k {\displaystyle k} -th power sum p k {\displaystyle p_{k}} of the roots of the characteristic polynomial of A {\displaystyle \mathbf {A} } , is given by its trace: p k = tr ( A k ) . {\displaystyle p_{k}=\operatorname {tr} (\mathbf {A} ^{k})\,.} The Newton identities now relate the traces of the powers A k {\displaystyle \mathbf {A} ^{k}} to the coefficients of the characteristic polynomial of A {\displaystyle \mathbf {A} } . Using them in reverse to express the elementary symmetric polynomials in terms of the power sums, they can be used to find the characteristic polynomial by computing only the powers A k {\displaystyle \mathbf {A} ^{k}} and their traces. This computation requires computing the traces of matrix powers A k {\displaystyle \mathbf {A} ^{k}} and solving a triangular system of equations. Both can be done in complexity class NC (solving a triangular system can be done by divide-and-conquer). Therefore, characteristic polynomial of a matrix can be computed in NC. By the Cayley–Hamilton theorem, every matrix satisfies its characteristic polynomial, and a simple transformation allows to find the adjugate matrix in NC. Rearranging the computations into an efficient form leads to the Faddeev–LeVerrier algorithm (1840), a fast parallel implementation of it is due to L. Csanky (1976). Its disadvantage is that it requires division by integers, so in general the field should have characteristic 0. === Relation with Galois theory === For a given n, the elementary symmetric polynomials ek(x1,...,xn) for k = 1,..., n form an algebraic basis for the space of symmetric polynomials in x1,.... xn: every polynomial expression in the xi that is invariant under all permutations of those variables is given by a polynomial expression in those elementary symmetric polynomials, and this expression is unique up to equivalence of polynomial expressions. This is a general fact known as the fundamental theorem of symmetric polynomials, and Newton's identities provide explicit formulae in the case of power sum symmetric polynomials. Applied to the monic polynomial t n + ∑ k = 1 n ( − 1 ) k a k t n − k {\textstyle t^{n}+\sum _{k=1}^{n}(-1)^{k}a_{k}t^{n-k}} with all coefficients ak considered as free parameters, this means that every symmetric polynomial expression S(x1,...,xn) in its roots can be expressed instead as a polynomial expression P(a1,...,an) in terms of its coefficients only, in other words without requiring knowledge of the roots. This fact also follows from general considerations in Galois theory (one views the ak as elements of a base field with roots in an extension field whose Galois group permutes them according to the full symmetric group, and the field fixed under all elements of the Galois group is the base field). The Newton identities also permit expressing the elementary symmetric polynomials in terms of the power sum symmetric polynomials, showing that any symmetric polynomial can also be expressed in the power sums. In fact the first n power sums also form an algebraic basis for the space of symmetric polynomials. == Related identities == There are a number of (families of) identities that, while they should be distinguished from Newton's identities, are very closely related to them. === A variant using complete homogeneous symmetric polynomials === Denoting by hk the complete homogeneous symmetric polynomial (that is, the sum of all monomials of degree k), the power sum polynomials also satisfy identities similar to Newton's identities, but not involving any minus signs. Expressed as identities of in the ring of symmetric functions, they read k h k = ∑ i = 1 k h k − i p i , {\displaystyle kh_{k}=\sum _{i=1}^{k}h_{k-i}p_{i},} valid for all n ≥ k ≥ 1. Contrary to Newton's identities, the left-hand sides do not become zero for large k, and the right-hand sides contain ever more non-zero terms. For the first few values of k, one has h 1 = p 1 , 2 h 2 = h 1 p 1 + p 2 , 3 h 3 = h 2 p 1 + h 1 p 2 + p 3 . {\displaystyle {\begin{aligned}h_{1}&=p_{1},\\2h_{2}&=h_{1}p_{1}+p_{2},\\3h_{3}&=h_{2}p_{1}+h_{1}p_{2}+p_{3}.\\\end{aligned}}} These relations can be justified by an argument analogous to the one by comparing coefficients in power series given above, based in this case on the generating function identity ∑ k = 0 ∞ h k ( x 1 , … , x n ) t k = ∏ i = 1 n 1 1 − x i t . {\displaystyle \sum _{k=0}^{\infty }h_{k}(x_{1},\ldots ,x_{n})t^{k}=\prod _{i=1}^{n}{\frac {1}{1-x_{i}t}}.} Proofs of Newton's identities, like these given below, cannot be easily adapted to prove these variants of those identities. === Expressing elementary symmetric polynomials in terms of power sums === As mentioned, Newton's identities can be used to recursively express elementary symmetric polynomials in terms of power sums. Doing so requires the introduction of integer denominators, so it can be done in the ring ΛQ of symmetric functions with rational coefficients: e 1 = p 1 , e 2 = 1 2 p 1 2 − 1 2 p 2 = 1 2 ( p 1 2 − p 2 ) , e 3 = 1 6 p 1 3 − 1 2 p 1 p 2 + 1 3 p 3 = 1 6 ( p 1 3 − 3 p 1 p 2 + 2 p 3 ) , e 4 = 1 24 p 1 4 − 1 4 p 1 2 p 2 + 1 8 p 2 2 + 1 3 p 1 p 3 − 1 4 p 4 = 1 24 ( p 1 4 − 6 p 1 2 p 2 + 3 p 2 2 + 8 p 1 p 3 − 6 p 4 ) , ⋮ e n = ( − 1 ) n ∑ m 1 + 2 m 2 + ⋯ + n m n = n m 1 ≥ 0 , … , m n ≥ 0 ∏ i = 1 n ( − p i ) m i m i ! i m i {\displaystyle {\begin{aligned}e_{1}&=p_{1},\\e_{2}&=\textstyle {\frac {1}{2}}p_{1}^{2}-{\frac {1}{2}}p_{2}&&=\textstyle {\frac {1}{2}}(p_{1}^{2}-p_{2}),\\e_{3}&=\textstyle {\frac {1}{6}}p_{1}^{3}-{\frac {1}{2}}p_{1}p_{2}+{\frac {1}{3}}p_{3}&&=\textstyle {\frac {1}{6}}(p_{1}^{3}-3p_{1}p_{2}+2p_{3}),\\e_{4}&=\textstyle {\frac {1}{24}}p_{1}^{4}-{\frac {1}{4}}p_{1}^{2}p_{2}+{\frac {1}{8}}p_{2}^{2}+{\frac {1}{3}}p_{1}p_{3}-{\frac {1}{4}}p_{4}&&=\textstyle {\frac {1}{24}}(p_{1}^{4}-6p_{1}^{2}p_{2}+3p_{2}^{2}+8p_{1}p_{3}-6p_{4}),\\&~~\vdots \\e_{n}&=(-1)^{n}\sum _{m_{1}+2m_{2}+\cdots +nm_{n}=n \atop m_{1}\geq 0,\ldots ,m_{n}\geq 0}\prod _{i=1}^{n}{\frac {(-p_{i})^{m_{i}}}{m_{i}!\,i^{m_{i}}}}\\\end{aligned}}} and so forth. The general formula can be conveniently expressed as e k = ( − 1 ) k k ! B k ( − p 1 , − 1 ! p 2 , − 2 ! p 3 , … , − ( k − 1 ) ! p k ) , {\displaystyle e_{k}={\frac {(-1)^{k}}{k!}}B_{k}(-p_{1},-1!\,p_{2},-2!\,p_{3},\ldots ,-(k-1)!\,p_{k}),} where the Bn is the complete exponential Bell polynomial. This expression also leads to the following identity for generating functions: ∑ k = 0 ∞ e k t k = exp ( ∑ k = 1 ∞ ( − 1 ) k + 1 k p k t k ) . {\displaystyle \sum _{k=0}^{\infty }e_{k}\,t^{k}=\exp \left(\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}p_{k}\,t^{k}\right).} Applied to a monic polynomial, these formulae express the coefficients in terms of the power sums of the roots: replace each ei by ai and each pk by sk. === Expressing complete homogeneous symmetric polynomials in terms of power sums === The analogous relations involving complete homogeneous symmetric polynomials can be similarly developed, giving equations h 1 = p 1 , h 2 = 1 2 p 1 2 + 1 2 p 2 = 1 2 ( p 1 2 + p 2 ) , h 3 = 1 6 p 1 3 + 1 2 p 1 p 2 + 1 3 p 3 = 1 6 ( p 1 3 + 3 p 1 p 2 + 2 p 3 ) , h 4 = 1 24 p 1 4 + 1 4 p 1 2 p 2 + 1 8 p 2 2 + 1 3 p 1 p 3 + 1 4 p 4 = 1 24 ( p 1 4 + 6 p 1 2 p 2 + 3 p 2 2 + 8 p 1 p 3 + 6 p 4 ) , ⋮ h k = ∑ m 1 + 2 m 2 + ⋯ + k m k = k m 1 ≥ 0 , … , m k ≥ 0 ∏ i = 1 k p i m i m i ! i m i {\displaystyle {\begin{aligned}h_{1}&=p_{1},\\h_{2}&=\textstyle {\frac {1}{2}}p_{1}^{2}+{\frac {1}{2}}p_{2}&&=\textstyle {\frac {1}{2}}(p_{1}^{2}+p_{2}),\\h_{3}&=\textstyle {\frac {1}{6}}p_{1}^{3}+{\frac {1}{2}}p_{1}p_{2}+{\frac {1}{3}}p_{3}&&=\textstyle {\frac {1}{6}}(p_{1}^{3}+3p_{1}p_{2}+2p_{3}),\\h_{4}&=\textstyle {\frac {1}{24}}p_{1}^{4}+{\frac {1}{4}}p_{1}^{2}p_{2}+{\frac {1}{8}}p_{2}^{2}+{\frac {1}{3}}p_{1}p_{3}+{\frac {1}{4}}p_{4}&&=\textstyle {\frac {1}{24}}(p_{1}^{4}+6p_{1}^{2}p_{2}+3p_{2}^{2}+8p_{1}p_{3}+6p_{4}),\\&~~\vdots \\h_{k}&=\sum _{m_{1}+2m_{2}+\cdots +km_{k}=k \atop m_{1}\geq 0,\ldots ,m_{k}\geq 0}\prod _{i=1}^{k}{\frac {p_{i}^{m_{i}}}{m_{i}!\,i^{m_{i}}}}\end{aligned}}} and so forth, in which there are only plus signs. In terms of the complete Bell polynomial, h k = 1 k ! B k ( p 1 , 0 ! p 2 , 1 ! p 3 , … , ( k − 1 ) ! p k ) . {\displaystyle h_{k}={\frac {1}{k!}}B_{k}(p_{1},0!\,p_{2},1!\,p_{3},\ldots ,(k-1)!\,p_{k}).} These expressions correspond exactly to the cycle index polynomials of the symmetric groups, if one interprets the power sums pi as indeterminates: the coefficient in the expression for hk of any monomial p1m1p2m2...plml is equal to the fraction of all permutations of k that have m1 fixed points, m2 cycles of length 2, ..., and ml cycles of length l. Explicitly, this coefficient can be written as 1 / N {\displaystyle 1/N} where N = ∏ i = 1 l ( m i ! i m i ) {\textstyle N=\prod _{i=1}^{l}(m_{i}!\,i^{m_{i}})} ; this N is the number permutations commuting with any given permutation π of the given cycle type. The expressions for the elementary symmetric functions have coefficients with the same absolute value, but a sign equal to the sign of π, namely (−1)m2+m4+.... It can be proved by considering the following inductive step: m f ( m ; m 1 , … , m n ) = f ( m − 1 ; m 1 − 1 , … , m n ) + ⋯ + f ( m − n ; m 1 , … , m n − 1 ) m 1 ∏ i = 1 n 1 i m i m i ! + ⋯ + n m n ∏ i = 1 n 1 i m i m i ! = m ∏ i = 1 n 1 i m i m i ! {\displaystyle {\begin{aligned}mf(m;m_{1},\ldots ,m_{n})&=f(m-1;m_{1}-1,\ldots ,m_{n})+\cdots +f(m-n;m_{1},\ldots ,m_{n}-1)\\m_{1}\prod _{i=1}^{n}{\frac {1}{i^{m_{i}}m_{i}!}}+\cdots +nm_{n}\prod _{i=1}^{n}{\frac {1}{i^{m_{i}}m_{i}!}}&=m\prod _{i=1}^{n}{\frac {1}{i^{m_{i}}m_{i}!}}\end{aligned}}} By analogy with the derivation of the generating function of the e n {\displaystyle e_{n}} , we can also obtain the generating function of the h n {\displaystyle h_{n}} , in terms of the power sums, as: ∑ k = 0 ∞ h k t k = exp ( ∑ k = 1 ∞ p k k t k ) . {\displaystyle \sum _{k=0}^{\infty }h_{k}\,t^{k}=\exp \left(\sum _{k=1}^{\infty }{\frac {p_{k}}{k}}\,t^{k}\right).} This generating function is thus the plethystic exponential of p 1 t = ( x 1 + ⋯ + x n ) t {\displaystyle p_{1}t=(x_{1}+\cdots +x_{n})t} . === Expressing power sums in terms of elementary symmetric polynomials === One may also use Newton's identities to express power sums in terms of elementary symmetric polynomials, which does not introduce denominators: p 1 = e 1 , p 2 = e 1 2 − 2 e 2 , p 3 = e 1 3 − 3 e 2 e 1 + 3 e 3 , p 4 = e 1 4 − 4 e 2 e 1 2 + 4 e 3 e 1 + 2 e 2 2 − 4 e 4 , p 5 = e 1 5 − 5 e 2 e 1 3 + 5 e 3 e 1 2 + 5 e 2 2 e 1 − 5 e 4 e 1 − 5 e 3 e 2 + 5 e 5 , p 6 = e 1 6 − 6 e 2 e 1 4 + 6 e 3 e 1 3 + 9 e 2 2 e 1 2 − 6 e 4 e 1 2 − 12 e 3 e 2 e 1 + 6 e 5 e 1 − 2 e 2 3 + 3 e 3 2 + 6 e 4 e 2 − 6 e 6 . {\displaystyle {\begin{aligned}p_{1}&=e_{1},\\p_{2}&=e_{1}^{2}-2e_{2},\\p_{3}&=e_{1}^{3}-3e_{2}e_{1}+3e_{3},\\p_{4}&=e_{1}^{4}-4e_{2}e_{1}^{2}+4e_{3}e_{1}+2e_{2}^{2}-4e_{4},\\p_{5}&=e_{1}^{5}-5e_{2}e_{1}^{3}+5e_{3}e_{1}^{2}+5e_{2}^{2}e_{1}-5e_{4}e_{1}-5e_{3}e_{2}+5e_{5},\\p_{6}&=e_{1}^{6}-6e_{2}e_{1}^{4}+6e_{3}e_{1}^{3}+9e_{2}^{2}e_{1}^{2}-6e_{4}e_{1}^{2}-12e_{3}e_{2}e_{1}+6e_{5}e_{1}-2e_{2}^{3}+3e_{3}^{2}+6e_{4}e_{2}-6e_{6}.\end{aligned}}} The first four formulas were obtained by Albert Girard in 1629 (thus before Newton). The general formula (for all positive integers m) is: p m = ( − 1 ) m m ∑ r 1 + 2 r 2 + ⋯ + m r m = m r 1 ≥ 0 , … , r m ≥ 0 ( r 1 + r 2 + ⋯ + r m − 1 ) ! r 1 ! r 2 ! ⋯ r m ! ∏ i = 1 m ( − e i ) r i . {\displaystyle p_{m}=(-1)^{m}m\sum _{r_{1}+2r_{2}+\cdots +mr_{m}=m \atop r_{1}\geq 0,\ldots ,r_{m}\geq 0}{\frac {(r_{1}+r_{2}+\cdots +r_{m}-1)!}{r_{1}!\,r_{2}!\cdots r_{m}!}}\prod _{i=1}^{m}(-e_{i})^{r_{i}}.} This can be conveniently stated in terms of ordinary Bell polynomials as p m = ( − 1 ) m m ∑ k = 1 m 1 k B ^ m , k ( − e 1 , … , − e m − k + 1 ) , {\displaystyle p_{m}=(-1)^{m}m\sum _{k=1}^{m}{\frac {1}{k}}{\hat {B}}_{m,k}(-e_{1},\ldots ,-e_{m-k+1}),} or equivalently as the generating function: ∑ k = 1 ∞ ( − 1 ) k − 1 p k t k k = ln ( 1 + e 1 t + e 2 t 2 + e 3 t 3 + ⋯ ) = e 1 t − 1 2 ( e 1 2 − 2 e 2 ) t 2 + 1 3 ( e 1 3 − 3 e 1 e 2 + 3 e 3 ) t 3 + ⋯ , {\displaystyle {\begin{aligned}\sum _{k=1}^{\infty }(-1)^{k-1}p_{k}{\frac {t^{k}}{k}}&=\ln \left(1+e_{1}t+e_{2}t^{2}+e_{3}t^{3}+\cdots \right)\\&=e_{1}t-{\frac {1}{2}}\left(e_{1}^{2}-2e_{2}\right)t^{2}+{\frac {1}{3}}\left(e_{1}^{3}-3e_{1}e_{2}+3e_{3}\right)t^{3}+\cdots ,\end{aligned}}} which is analogous to the Bell polynomial exponential generating function given in the previous subsection. The multiple summation formula above can be proved by considering the following inductive step: f ( m ; r 1 , … , r n ) = f ( m − 1 ; r 1 − 1 , ⋯ , r n ) + ⋯ + f ( m − n ; r 1 , … , r n − 1 ) = 1 ( r 1 − 1 ) ! ⋯ r n ! ( m − 1 ) ( r 1 + ⋯ + r n − 2 ) ! + ⋯ ⋯ + 1 r 1 ! ⋯ ( r n − 1 ) ! ( m − n ) ( r 1 + ⋯ + r n − 2 ) ! = 1 r 1 ! ⋯ r n ! [ r 1 ( m − 1 ) + ⋯ + r n ( m − n ) ] [ r 1 + ⋯ + r n − 2 ] ! = 1 r 1 ! ⋯ r n ! [ m ( r 1 + ⋯ + r n ) − m ] [ r 1 + ⋯ + r n − 2 ] ! = m ( r 1 + ⋯ + r n − 1 ) ! r 1 ! ⋯ r n ! {\displaystyle {\begin{aligned}f(m;\;r_{1},\ldots ,r_{n})={}&f(m-1;\;r_{1}-1,\cdots ,r_{n})+\cdots +f(m-n;\;r_{1},\ldots ,r_{n}-1)\\[8pt]={}&{\frac {1}{(r_{1}-1)!\cdots r_{n}!}}(m-1)(r_{1}+\cdots +r_{n}-2)!+\cdots \\&\cdots +{\frac {1}{r_{1}!\cdots (r_{n}-1)!}}(m-n)(r_{1}+\cdots +r_{n}-2)!\\[8pt]={}&{\frac {1}{r_{1}!\cdots r_{n}!}}\left[r_{1}(m-1)+\cdots +r_{n}(m-n)\right]\left[r_{1}+\cdots +r_{n}-2\right]!\\[8pt]={}&{\frac {1}{r_{1}!\cdots r_{n}!}}\left[m(r_{1}+\cdots +r_{n})-m\right]\left[r_{1}+\cdots +r_{n}-2\right]!\\[8pt]={}&{\frac {m(r_{1}+\cdots +r_{n}-1)!}{r_{1}!\cdots r_{n}!}}\end{aligned}}} === Expressing power sums in terms of complete homogeneous symmetric polynomials === Finally one may use the variant identities involving complete homogeneous symmetric polynomials similarly to express power sums in term of them: p 1 = + h 1 , p 2 = − h 1 2 + 2 h 2 , p 3 = + h 1 3 − 3 h 2 h 1 + 3 h 3 , p 4 = − h 1 4 + 4 h 2 h 1 2 − 4 h 3 h 1 − 2 h 2 2 + 4 h 4 , p 5 = + h 1 5 − 5 h 2 h 1 3 + 5 h 2 2 h 1 + 5 h 3 h 1 2 − 5 h 3 h 2 − 5 h 4 h 1 + 5 h 5 , p 6 = − h 1 6 + 6 h 2 h 1 4 − 9 h 2 2 h 1 2 − 6 h 3 h 1 3 + 2 h 2 3 + 12 h 3 h 2 h 1 + 6 h 4 h 1 2 − 3 h 3 2 − 6 h 4 h 2 − 6 h 1 h 5 + 6 h 6 , {\displaystyle {\begin{aligned}p_{1}&=+h_{1},\\p_{2}&=-h_{1}^{2}+2h_{2},\\p_{3}&=+h_{1}^{3}-3h_{2}h_{1}+3h_{3},\\p_{4}&=-h_{1}^{4}+4h_{2}h_{1}^{2}-4h_{3}h_{1}-2h_{2}^{2}+4h_{4},\\p_{5}&=+h_{1}^{5}-5h_{2}h_{1}^{3}+5h_{2}^{2}h_{1}+5h_{3}h_{1}^{2}-5h_{3}h_{2}-5h_{4}h_{1}+5h_{5},\\p_{6}&=-h_{1}^{6}+6h_{2}h_{1}^{4}-9h_{2}^{2}h_{1}^{2}-6h_{3}h_{1}^{3}+2h_{2}^{3}+12h_{3}h_{2}h_{1}+6h_{4}h_{1}^{2}-3h_{3}^{2}-6h_{4}h_{2}-6h_{1}h_{5}+6h_{6},\\\end{aligned}}} and so on. Apart from the replacement of each ei by the corresponding hi, the only change with respect to the previous family of identities is in the signs of the terms, which in this case depend just on the number of factors present: the sign of the monomial ∏ i = 1 l h i m i {\textstyle \prod _{i=1}^{l}h_{i}^{m_{i}}} is −(−1)m1+m2+m3+.... In particular the above description of the absolute value of the coefficients applies here as well. The general formula (for all non-negative integers m) is: p m = − ∑ r 1 + 2 r 2 + ⋯ + m r m = m r 1 ≥ 0 , … , r m ≥ 0 m ( r 1 + r 2 + ⋯ + r m − 1 ) ! r 1 ! r 2 ! ⋯ r m ! ∏ i = 1 m ( − h i ) r i {\displaystyle p_{m}=-\sum _{r_{1}+2r_{2}+\cdots +mr_{m}=m \atop r_{1}\geq 0,\ldots ,r_{m}\geq 0}{\frac {m(r_{1}+r_{2}+\cdots +r_{m}-1)!}{r_{1}!\,r_{2}!\cdots r_{m}!}}\prod _{i=1}^{m}(-h_{i})^{r_{i}}} === Expressions as determinants === One can obtain explicit formulas for the above expressions in the form of determinants, by considering the first n of Newton's identities (or it counterparts for the complete homogeneous polynomials) as linear equations in which the elementary symmetric functions are known and the power sums are unknowns (or vice versa), and apply Cramer's rule to find the solution for the final unknown. For instance taking Newton's identities in the form e 1 = 1 p 1 , 2 e 2 = e 1 p 1 − 1 p 2 , 3 e 3 = e 2 p 1 − e 1 p 2 + 1 p 3 , ⋮ n e n = e n − 1 p 1 − e n − 2 p 2 + ⋯ + ( − 1 ) n e 1 p n − 1 + ( − 1 ) n − 1 p n {\displaystyle {\begin{aligned}e_{1}&=1p_{1},\\2e_{2}&=e_{1}p_{1}-1p_{2},\\3e_{3}&=e_{2}p_{1}-e_{1}p_{2}+1p_{3},\\&\,\,\,\vdots \\ne_{n}&=e_{n-1}p_{1}-e_{n-2}p_{2}+\cdots +(-1)^{n}e_{1}p_{n-1}+(-1)^{n-1}p_{n}\end{aligned}}} we consider p 1 , − p 2 , p 3 , … , ( − 1 ) n p n − 1 {\displaystyle p_{1},-p_{2},p_{3},\ldots ,(-1)^{n}p_{n-1}} and p n {\displaystyle p_{n}} as unknowns, and solve for the final one, giving p n = | 1 0 ⋯ e 1 e 1 1 0 ⋯ 2 e 2 e 2 e 1 1 3 e 3 ⋮ ⋱ ⋱ ⋮ e n − 1 ⋯ e 2 e 1 n e n | | 1 0 ⋯ e 1 1 0 ⋯ e 2 e 1 1 ⋮ ⋱ ⋱ e n − 1 ⋯ e 2 e 1 1 | − 1 = ( − 1 ) n − 1 | 1 0 ⋯ e 1 e 1 1 0 ⋯ 2 e 2 e 2 e 1 1 3 e 3 ⋮ ⋱ ⋱ ⋮ e n − 1 ⋯ e 2 e 1 n e n | = | e 1 1 0 ⋯ 2 e 2 e 1 1 0 ⋯ 3 e 3 e 2 e 1 1 ⋮ ⋱ ⋱ n e n e n − 1 ⋯ e 1 | . {\displaystyle {\begin{aligned}p_{n}={}&{\begin{vmatrix}1&0&\cdots &&e_{1}\\e_{1}&1&0&\cdots &2e_{2}\\e_{2}&e_{1}&1&&3e_{3}\\\vdots &&\ddots &\ddots &\vdots \\e_{n-1}&\cdots &e_{2}&e_{1}&ne_{n}\end{vmatrix}}{\begin{vmatrix}1&0&\cdots &\\e_{1}&1&0&\cdots \\e_{2}&e_{1}&1&\\\vdots &&\ddots &\ddots \\e_{n-1}&\cdots &e_{2}&e_{1}&1\end{vmatrix}}^{-1}\\[7pt]={(-1)^{n-1}}&{\begin{vmatrix}1&0&\cdots &&e_{1}\\e_{1}&1&0&\cdots &2e_{2}\\e_{2}&e_{1}&1&&3e_{3}\\\vdots &&\ddots &\ddots &\vdots \\e_{n-1}&\cdots &e_{2}&e_{1}&ne_{n}\end{vmatrix}}\\[7pt]={}&{\begin{vmatrix}e_{1}&1&0&\cdots \\2e_{2}&e_{1}&1&0&\cdots \\3e_{3}&e_{2}&e_{1}&1\\\vdots &&&\ddots &\ddots \\ne_{n}&e_{n-1}&\cdots &&e_{1}\end{vmatrix}}.\end{aligned}}} Solving for e n {\displaystyle e_{n}} instead of for p n {\displaystyle p_{n}} is similar, as the analogous computations for the complete homogeneous symmetric polynomials; in each case the details are slightly messier than the final results, which are (Macdonald 1979, p. 20): e n = 1 n ! | p 1 1 0 ⋯ p 2 p 1 2 0 ⋯ ⋮ ⋱ ⋱ p n − 1 p n − 2 ⋯ p 1 n − 1 p n p n − 1 ⋯ p 2 p 1 | p n = ( − 1 ) n − 1 | h 1 1 0 ⋯ 2 h 2 h 1 1 0 ⋯ 3 h 3 h 2 h 1 1 ⋮ ⋱ ⋱ n h n h n − 1 ⋯ h 1 | h n = 1 n ! | p 1 − 1 0 ⋯ p 2 p 1 − 2 0 ⋯ ⋮ ⋱ ⋱ p n − 1 p n − 2 ⋯ p 1 1 − n p n p n − 1 ⋯ p 2 p 1 | . {\displaystyle {\begin{aligned}e_{n}={\frac {1}{n!}}&{\begin{vmatrix}p_{1}&1&0&\cdots \\p_{2}&p_{1}&2&0&\cdots \\\vdots &&\ddots &\ddots \\p_{n-1}&p_{n-2}&\cdots &p_{1}&n-1\\p_{n}&p_{n-1}&\cdots &p_{2}&p_{1}\end{vmatrix}}\\[7pt]p_{n}=(-1)^{n-1}&{\begin{vmatrix}h_{1}&1&0&\cdots \\2h_{2}&h_{1}&1&0&\cdots \\3h_{3}&h_{2}&h_{1}&1\\\vdots &&&\ddots &\ddots \\nh_{n}&h_{n-1}&\cdots &&h_{1}\end{vmatrix}}\\[7pt]h_{n}={\frac {1}{n!}}&{\begin{vmatrix}p_{1}&-1&0&\cdots \\p_{2}&p_{1}&-2&0&\cdots \\\vdots &&\ddots &\ddots \\p_{n-1}&p_{n-2}&\cdots &p_{1}&1-n\\p_{n}&p_{n-1}&\cdots &p_{2}&p_{1}\end{vmatrix}}.\end{aligned}}} Note that the use of determinants makes that the formula for h n {\displaystyle h_{n}} has additional minus signs compared to the one for e n {\displaystyle e_{n}} , while the situation for the expanded form given earlier is opposite. As remarked in (Littlewood 1950, p. 84) one can alternatively obtain the formula for h n {\displaystyle h_{n}} by taking the permanent of the matrix for e n {\displaystyle e_{n}} instead of the determinant, and more generally an expression for any Schur polynomial can be obtained by taking the corresponding immanant of this matrix. == Derivation of the identities == Each of Newton's identities can easily be checked by elementary algebra; however, their validity in general needs a proof. Here are some possible derivations. === From the special case n = k === One can obtain the k-th Newton identity in k variables by substitution into ∏ i = 1 k ( t − x i ) = ∑ i = 0 k ( − 1 ) k − i e k − i ( x 1 , … , x k ) t i {\displaystyle \prod _{i=1}^{k}(t-x_{i})=\sum _{i=0}^{k}(-1)^{k-i}e_{k-i}(x_{1},\ldots ,x_{k})t^{i}} as follows. Substituting xj for t gives 0 = ∑ i = 0 k ( − 1 ) k − i e k − i ( x 1 , … , x k ) x j i for 1 ≤ j ≤ k {\displaystyle 0=\sum _{i=0}^{k}(-1)^{k-i}e_{k-i}(x_{1},\ldots ,x_{k}){x_{j}}^{i}\quad {\text{for }}1\leq j\leq k} Summing over all j gives 0 = ( − 1 ) k k e k ( x 1 , … , x k ) + ∑ i = 1 k ( − 1 ) k − i e k − i ( x 1 , … , x k ) p i ( x 1 , … , x k ) , {\displaystyle 0=(-1)^{k}ke_{k}(x_{1},\ldots ,x_{k})+\sum _{i=1}^{k}(-1)^{k-i}e_{k-i}(x_{1},\ldots ,x_{k})p_{i}(x_{1},\ldots ,x_{k}),} where the terms for i = 0 were taken out of the sum because p0 is (usually) not defined. This equation immediately gives the k-th Newton identity in k variables. Since this is an identity of symmetric polynomials (homogeneous) of degree k, its validity for any number of variables follows from its validity for k variables. Concretely, the identities in n < k variables can be deduced by setting k − n variables to zero. The k-th Newton identity in n > k variables contains more terms on both sides of the equation than the one in k variables, but its validity will be assured if the coefficients of any monomial match. Because no individual monomial involves more than k of the variables, the monomial will survive the substitution of zero for some set of n − k (other) variables, after which the equality of coefficients is one that arises in the k-th Newton identity in k (suitably chosen) variables. === Comparing coefficients in series === Another derivation can be obtained by computations in the ring of formal power series R[[t]], where R is Z[x1,..., xn], the ring of polynomials in n variables x1,..., xn over the integers. Starting again from the basic relation ∏ i = 1 n ( t − x i ) = ∑ k = 0 n ( − 1 ) k a k t n − k {\displaystyle \prod _{i=1}^{n}(t-x_{i})=\sum _{k=0}^{n}(-1)^{k}a_{k}t^{n-k}} and "reversing the polynomials" by substituting 1/t for t and then multiplying both sides by tn to remove negative powers of t, gives ∏ i = 1 n ( 1 − x i t ) = ∑ k = 0 n ( − 1 ) k a k t k . {\displaystyle \prod _{i=1}^{n}(1-x_{i}t)=\sum _{k=0}^{n}(-1)^{k}a_{k}t^{k}.} (the above computation should be performed in the field of fractions of R[[t]]; alternatively, the identity can be obtained simply by evaluating the product on the left side) Swapping sides and expressing the ai as the elementary symmetric polynomials they stand for gives the identity ∑ k = 0 n ( − 1 ) k e k ( x 1 , … , x n ) t k = ∏ i = 1 n ( 1 − x i t ) . {\displaystyle \sum _{k=0}^{n}(-1)^{k}e_{k}(x_{1},\ldots ,x_{n})t^{k}=\prod _{i=1}^{n}(1-x_{i}t).} One formally differentiates both sides with respect to t, and then (for convenience) multiplies by t, to obtain ∑ k = 0 n ( − 1 ) k k e k ( x 1 , … , x n ) t k = t ∑ i = 1 n [ ( − x i ) ∏ j ≠ i ( 1 − x j t ) ] = − ( ∑ i = 1 n x i t 1 − x i t ) ∏ j = 1 n ( 1 − x j t ) = − [ ∑ i = 1 n ∑ j = 1 ∞ ( x i t ) j ] [ ∑ ℓ = 0 n ( − 1 ) ℓ e ℓ ( x 1 , … , x n ) t ℓ ] = [ ∑ j = 1 ∞ p j ( x 1 , … , x n ) t j ] [ ∑ ℓ = 0 n ( − 1 ) ℓ − 1 e ℓ ( x 1 , … , x n ) t ℓ ] , {\displaystyle {\begin{aligned}\sum _{k=0}^{n}(-1)^{k}ke_{k}(x_{1},\ldots ,x_{n})t^{k}&=t\sum _{i=1}^{n}\left[(-x_{i})\prod \nolimits _{j\neq i}(1-x_{j}t)\right]\\&=-\left(\sum _{i=1}^{n}{\frac {x_{i}t}{1-x_{i}t}}\right)\prod \nolimits _{j=1}^{n}(1-x_{j}t)\\&=-\left[\sum _{i=1}^{n}\sum _{j=1}^{\infty }(x_{i}t)^{j}\right]\left[\sum _{\ell =0}^{n}(-1)^{\ell }e_{\ell }(x_{1},\ldots ,x_{n})t^{\ell }\right]\\&=\left[\sum _{j=1}^{\infty }p_{j}(x_{1},\ldots ,x_{n})t^{j}\right]\left[\sum _{\ell =0}^{n}(-1)^{\ell -1}e_{\ell }(x_{1},\ldots ,x_{n})t^{\ell }\right],\\\end{aligned}}} where the polynomial on the right hand side was first rewritten as a rational function in order to be able to factor out a product out of the summation, then the fraction in the summand was developed as a series in t, using the formula X 1 − X = X + X 2 + X 3 + X 4 + X 5 + ⋯ , {\displaystyle {\frac {X}{1-X}}=X+X^{2}+X^{3}+X^{4}+X^{5}+\cdots ,} and finally the coefficient of each t j was collected, giving a power sum. (The series in t is a formal power series, but may alternatively be thought of as a series expansion for t sufficiently close to 0, for those more comfortable with that; in fact one is not interested in the function here, but only in the coefficients of the series.) Comparing coefficients of tk on both sides one obtains ( − 1 ) k k e k ( x 1 , … , x n ) = ∑ j = 1 k ( − 1 ) k − j − 1 p j ( x 1 , … , x n ) e k − j ( x 1 , … , x n ) , {\displaystyle (-1)^{k}ke_{k}(x_{1},\ldots ,x_{n})=\sum _{j=1}^{k}(-1)^{k-j-1}p_{j}(x_{1},\ldots ,x_{n})e_{k-j}(x_{1},\ldots ,x_{n}),} which gives the k-th Newton identity. === As a telescopic sum of symmetric function identities === The following derivation, given essentially in (Mead, 1992), is formulated in the ring of symmetric functions for clarity (all identities are independent of the number of variables). Fix some k > 0, and define the symmetric function r(i) for 2 ≤ i ≤ k as the sum of all distinct monomials of degree k obtained by multiplying one variable raised to the power i with k − i distinct other variables (this is the monomial symmetric function mγ where γ is a hook shape (i,1,1,...,1)). In particular r(k) = pk; for r(1) the description would amount to that of ek, but this case was excluded since here monomials no longer have any distinguished variable. All products piek−i can be expressed in terms of the r(j) with the first and last case being somewhat special. One has p i e k − i = r ( i ) + r ( i + 1 ) for 1 < i < k {\displaystyle p_{i}e_{k-i}=r(i)+r(i+1)\quad {\text{for }}1<i<k} since each product of terms on the left involving distinct variables contributes to r(i), while those where the variable from pi already occurs among the variables of the term from ek−i contributes to r(i + 1), and all terms on the right are so obtained exactly once. For i = k one multiplies by e0 = 1, giving trivially p k e 0 = p k = r ( k ) . {\displaystyle p_{k}e_{0}=p_{k}=r(k).} Finally the product p1ek−1 for i = 1 gives contributions to r(i + 1) = r(2) like for other values i < k, but the remaining contributions produce k times each monomial of ek, since any one of the variables may come from the factor p1; thus p 1 e k − 1 = k e k + r ( 2 ) . {\displaystyle p_{1}e_{k-1}=ke_{k}+r(2).} The k-th Newton identity is now obtained by taking the alternating sum of these equations, in which all terms of the form r(i) cancel out. === Combinatorial proof === A short combinatorial proof of Newton's identities was given by Doron Zeilberger in 1984. == See also == Power sum symmetric polynomial Elementary symmetric polynomial Newton's inequalities Symmetric function Fluid solutions, an article giving an application of Newton's identities to computing the characteristic polynomial of the Einstein tensor in the case of a perfect fluid, and similar articles on other types of exact solutions in general relativity. == References == Tignol, Jean-Pierre (2001). Galois' theory of algebraic equations. Singapore: World Scientific. ISBN 978-981-02-4541-2. Bergeron, F.; Labelle, G. & Leroux, P. (1998). Combinatorial species and tree-like structures. Cambridge: Cambridge University Press. ISBN 978-0-521-57323-8. Cameron, Peter J. (1999). Permutation Groups. Cambridge: Cambridge University Press. ISBN 978-0-521-65378-7. Cox, David; Little, John & O'Shea, Donal (1992). Ideals, Varieties, and Algorithms. New York: Springer-Verlag. ISBN 978-0-387-97847-5. Eppstein, D.; Goodrich, M. T. (2007). "Space-efficient straggler identification in round-trip data streams via Newton's identities and invertible Bloom filters". Algorithms and Data Structures, 10th International Workshop, WADS 2007. Springer-Verlag, Lecture Notes in Computer Science 4619. pp. 637–648. arXiv:0704.3313. Bibcode:2007arXiv0704.3313E. Littlewood, D. E. (1950). The theory of group characters and matrix representations of groups. Oxford: Oxford University Press. viii+310. ISBN 0-8218-4067-3. {{cite book}}: ISBN / Date incompatibility (help) Macdonald, I. G. (1979). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs. Oxford: The Clarendon Press, Oxford University Press. viii+180. ISBN 0-19-853530-9. MR 0553598. Macdonald, I. G. (1995). Symmetric functions and Hall polynomials. Oxford Mathematical Monographs (Second ed.). New York: Oxford Science Publications. The Clarendon Press, Oxford University Press. p. x+475. ISBN 0-19-853489-2. MR 1354144. Mead, D.G. (1992). "Newton's Identities". The American Mathematical Monthly. 99 (8). Mathematical Association of America: 749–751. doi:10.2307/2324242. JSTOR 2324242. Stanley, Richard P. (1999). Enumerative Combinatorics, Vol. 2. Cambridge University Press. ISBN 0-521-56069-1. (hardback). (paperback). Sturmfels, Bernd (1992). Algorithms in Invariant Theory. New York: Springer-Verlag. ISBN 978-0-387-82445-1. Tucker, Alan (1980). Applied Combinatorics (5/e ed.). New York: Wiley. ISBN 978-0-471-73507-6. == External links == Newton–Girard formulas on MathWorld A Matrix Proof of Newton's Identities in Mathematics Magazine Application on the number of real roots A Combinatorial Proof of Newton's Identities by Doron Zeilberger
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Wikipedia:Newton's inequalities#0
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In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1, a2, ..., an are non-negative real numbers and let e k {\displaystyle e_{k}} denote the kth elementary symmetric polynomial in a1, a2, ..., an. Then the elementary symmetric means, given by S k = e k ( n k ) , {\displaystyle S_{k}={\frac {e_{k}}{\binom {n}{k}}},} satisfy the inequality S k − 1 S k + 1 ≤ S k 2 . {\displaystyle S_{k-1}S_{k+1}\leq S_{k}^{2}.} Equality holds if and only if all the numbers ai are equal. It can be seen that S1 is the arithmetic mean, and Sn is the n-th power of the geometric mean. == See also == Maclaurin's inequality == References == Hardy, G. H.; Littlewood, J. E.; Pólya, G. (1952). Inequalities. Cambridge University Press. ISBN 978-0521358804. {{cite book}}: ISBN / Date incompatibility (help) Newton, Isaac (1707). Arithmetica universalis: sive de compositione et resolutione arithmetica liber. D.S. Bernstein Matrix Mathematics: Theory, Facts, and Formulas (2009 Princeton) p. 55 Maclaurin, C. (1729). "A second letter to Martin Folks, Esq.; concerning the roots of equations, with the demonstration of other rules in algebra". Philosophical Transactions. 36 (407–416): 59–96. doi:10.1098/rstl.1729.0011. Whiteley, J.N. (1969). "On Newton's Inequality for Real Polynomials". The American Mathematical Monthly. 76 (8). The American Mathematical Monthly, Vol. 76, No. 8: 905–909. doi:10.2307/2317943. JSTOR 2317943. Niculescu, Constantin (2000). "A New Look at Newton's Inequalities". Journal of Inequalities in Pure and Applied Mathematics. 1 (2). Article 17.
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Wikipedia:Ngamta Thamwattana#0
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Ngamta ""Natalie"" Thamwattana is a Thai mathematician who works in Australia as a Professor of Applied Mathematics at the University of Newcastle (Australia). In 2014 she won the J. H. Michell Medal of ANZIAM for her "pioneering contributions in the areas of granular materials and nanotechnology". Thamwattana came from Chumphon Province in Thailand, where her mother made dresses and her father worked in the police. After her first year of university, her high test scores in science and mathematics earned her a government scholarship with full tuition and, later, support for doctoral studies abroad. She earned a bachelor's degree in mathematics from Mahidol University in Thailand in 2000, choosing mathematics because as a mathematician she wouldn't "have to play with the chemical stuff and we didn't have to kill rat or any insects". She completed her Ph.D. at Wollongong in 2004 under the supervision of James Murray Hill. After completing her studies, she returned the money from her Thai scholarship in order to stay in Australia with her new husband, a lecturer at Wollongong. She joined the Wollongong faculty herself, and founded the Nanomechanics Group there. In 2018 she moved to University of Newcastle (Australia) to take up a position as professor of Applied Mathematics. == References == == External links == Ngamta Thamwattana publications indexed by Google Scholar
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Wikipedia:Nicholas A. M. Monk#0
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Nicholas A. M. Monk is a physicist and mathematician. He is Alexander von Humboldt Foundation German Research Chair at AIMS Ghana and Professor of Mathematical Biology at the University of Sheffield and the University of Ghana. He is known for his works on mathematical biology, pattern formation, and dynamical systems. He earned his PhD in 1994 from Birkbeck College, University of London. His advisor was Basil Hiley. == References == == External links == Personal website
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Wikipedia:Nicholas M. Smith Jr.#0
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Nicholas Monroe Smith Jr. (1914 – 2003) was a nuclear physicist and research consultant. Smith was an expert on reactor physics, a developer of operations research/computer modeling, and a computer applications consultant. He had ties to the Manhattan Project at Chicago and Oak Ridge, and worked with Samuel Allison and James Van Allen. Smith was a pioneer in the field of operations research. == Early life and education == Smith was born on March 23, 1914, in Little Rock, Arkansas, the son of Nick Monroe Smith and Mary Gossett. He attended the University of Arkansas and received his Bachelor of Arts degree in mathematics and physics. According to the US Census, in 1940 Smith and his wife Elizabeth resided in Chicago, Illinois. At the University of Chicago, he earned a master's and doctoral degrees in physics. He worked in the Ryerson Physical Laboratory, At University of Chicago, his advisor was Samuel Allison and graduate studies involved work on Chicago Pile-1, the first controlled nuclear chain reaction by Enrico Fermi. Smith landed a postdoctoral fellowship at the Carnegie Institution of Washington, Washington, D.C., and performed research with James Van Allen in the Department of Terrestrial Magnetism. In addition to Allison, Smith worked with physicist Lester Skaggs to design an aircraft proximity detection system that utilized radio waves to locate and detonate anti-aircraft shells. == Career as a physicist == Following the outbreak of World War II, Smith obtained a position at the Johns Hopkins University Applied Physics Laboratory in Maryland. As a civilian scientist, he was assigned to the Army Air Force in England, and planned railway targets for airstrikes in support of D-Day. For this work he was presented with the Medal of Freedom. After World War II, Smith worked as a physicist at Oak Ridge National Laboratory in Tennessee from 1946 to 1951. He studied and reported on the dangers of radioactive material contamination from nuclear weapons. In 1949, Smith at Oak Ridge conducted a study sponsored by the Atomic Energy Commission (A.E.C.)'s Division of Biology and Medicine, and performed calculations to determine the theoretical number of atomic bomb detonations necessary to achieve significant radiation exposure and radioactive material fallout. In 1951 after the Ranger and Greenhouse tests, Smith reassessed the earlier calculations and estimates. He determined that detonation of 100,000 Nagasaki type bombs would be sufficient to achieve the doomsday effect. With this information, the A.E.C.'s staff of the Division of Biology and Medicine concluded this to be extremely remote and dubbed the study as Project GABRIEL. == Project GABRIEL == In the AEC, the group responsible for Project GABRIEL was the Division of Biology and Medicine. The Division was charged with maintenance of experimental studies and field studies. The Division was required to collect and analyze data from internal and external sources. In 1949, Smith performed a theoretical analysis of the long term aspects of Project GABRIEL. He reached the conclusion that: Sr-90 is by far the most hazardous isotope resulting from nuclear detonations, and that the distribution of this isotope over large areas of the earth's surface constitutes the limiting factor in estimating the long-range hazard from the use of a large number of atomic bombs. In 1952, the RAND Corporation completed a study of Project GABRIEL, and was charged with analyzing the short term characteristics of nuclear fallout. The study was dubbed Project AUREOLE. == Operations research == For 20 years, Smith worked at Research Analysis Corporation as leader in the Advanced Research Department, a U.S. Army funded successor to the Operations Research Office. The focus of the work was war games simulation and nonlinear computer programming. His department produced numerous professional papers, including two Lanchester prize-winning books. In 1971, Smith founded TELIMIS Corporation, based in Springfield, Virginia, a company that developed applications in computer technology. He went on to work as a consultant and served as chief scientist at the Washington Institute of Technology in Fairfax, Virginia. == Death == Smith died on August 7, 2003, at his home in Lusby, Maryland, of metastatic prostate cancer. == Awards and honors == Medal of Freedom, for World War II work in operations research Phi Beta Kappa Sigma Xi == Professional affiliations == American Society for Cybernetics Society for General Systems Research Operations Research Society of America === Patents === Apparatus for observing the conduct of a projectile in a gun. Microwave measuring of projectile speed. == Death == Smith died August 7, 2003. == References ==
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Wikipedia:Nick Woodhouse#0
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The Honourable Nicholas Michael John Woodhouse (born 27 February 1949) is a British mathematician. He is Emeritus Fellow of Wadham College, University of Oxford and former President of the Clay Mathematics Institute. == Education and early life == Woodhouse is the younger son and second child of Christopher Montague Woodhouse, 5th Baron Terrington, and Lady Davidema (Davina) Bulwer-Lytton. He was educated at Winchester College, then the University of Oxford and King's College London. He completed his undergraduate degree in Mathematics at Christ Church, Oxford in 1970 and PhD at the University of London, King's College in 1973, where he subsequently held an SRC Fellowship. In 1973, he also held a Procter Visiting Fellowship at Princeton University. == Career == He was Research Associate in the Department of Physics, Princeton University (1975-1976) and a Junior Lecturer, University of Oxford (1976-1977). In 1977, he was made a Fellow of Wadham College, University of Oxford, where he has been an Emeritus Fellow since 1988. He was Head of the Mathematical Institute, University of Oxford (2001-2009). He served as Treasurer of the London Mathematical Society (2002-2009) and President of the Clay Mathematics Institute (2012-2018). His research is in applications of geometry in mathematical physics. == Honours == Woodhouse was appointed Commander of the Order of the British Empire (CBE) in the 2020 New Year Honours for services to mathematics. == Bibliography == Lectures on Geometric Quantization, with D J Simms, Lecture Notes in Physics, Springer, 1976 Introduction to Analytical Dynamics, OUP, 1987; Springer, 2010 Geometric Quantization, OUP, 1980 (2nd edition 1997) Special Relativity, Lecture Notes in Physics, Monographs, Springer, 1992 Integrability, Self-Duality, and Twistor Theory, with L.J. Mason, OUP/LMS, 1995 Special Relativity, Springer, 2004 General Relativity, Springer, 2007 == References == == External links == Official website
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Wikipedia:Nick Wormald#0
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Nicholas Charles Wormald (born 1953) is an Australian mathematician and professor of mathematics at Monash University. He specializes in probabilistic combinatorics, graph theory, graph algorithms, Steiner trees, web graphs, mine optimization, and other areas in combinatorics. In 1979, Wormald earned a Ph.D. in mathematics from the University of Newcastle with a dissertation titled Some problems in the enumeration of labelled graphs. In 2006, he won the Euler Medal from the Institute of Combinatorics and its Applications. He has held the Canada Research Chair in Combinatorics and Optimization at the University of Waterloo. In 2012, he was recognized with an Australian Laureate Fellowship for his achievements. In 2017, he was elected as a Fellow of the Australian Academy of Science. In 2018, Wormald was an invited speaker at the International Congress of Mathematicians in Rio de Janeiro. == Selected publications == Nicholas C. Wormald (1999). "Models of random regular graphs" (PDF). London Mathematical Society Lecture Note Series. Cambridge University Press: 239–298. Peter Eades; Nicholas C. Wormald (1994). "Edge crossings in drawings of bipartite graphs". Algorithmica. 11 (4). Springer: 379–403. doi:10.1007/BF01187020. S2CID 22476033. Nicholas C. Wormald (1995). "Differential equations for random processes and random graphs". Annals of Applied Probability. 5 (4). JSTOR: 1217–1235. doi:10.1214/aoap/1177004612. Nicholas C Wormald (1999). "The differential equation method for random graph processes and greedy algorithms" (PDF). Lectures on Approximation and Randomized Algorithms. Citeseer: 73–155. Robert W. Robinson; Nicholas C. Wormald (1994). "Almost all regular graphs are Hamiltonian". Random Structures & Algorithms. 5 (2). Wiley Online Library: 363–374. doi:10.1002/rsa.3240050209. Brendan D McKay; Nicholas C Wormald (1991). "Asymptotic enumeration by degree sequence of graphs with degrees o ( n ½ ) " (PDF). Combinatorica. 11 (4). Springer: 369–382. doi:10.1007/bf01275671. S2CID 9228526. Angelika Steger; Nicholas C. Wormald (1999). "Generating random regular graphs quickly". Combinatorics, Probability and Computing. 8 (4). Cambridge Univ Press: 377–396. doi:10.1017/S0963548399003867. S2CID 14545326. Nicholas C. Wormald (1981). "The asymptotic connectivity of labelled regular graphs". Journal of Combinatorial Theory. Series B. 31 (2). Elsevier: 156–167. doi:10.1016/S0095-8956(81)80021-4. == References ==
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Wikipedia:Nicolae Culianu#0
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Ioan Petru Culianu or Couliano (5 January 1950 – 21 May 1991) was a Romanian historian of religion, culture, and ideas, a philosopher and political essayist, and a short story writer. He served as professor of the history of religions at the University of Chicago from 1988 to his death, and had previously taught the history of Romanian culture at the University of Groningen. An expert in Gnosticism and Renaissance magic, he was encouraged and befriended by Mircea Eliade, though he gradually distanced himself from his mentor. Culianu published seminal work on the interrelation of the occult, Eros, magic, physics, and history. Culianu was murdered in 1991. It has been much speculated his murder was in consequence of his critical view of Romanian national politics. Some factions of the Romanian political right openly celebrated his murder. The Romanian Securitate, which he once lambasted as a force "of epochal stupidity", has also been suspected of involvement and of using puppet fronts on the right as cover. == Biography == === Education and career === Culianu was born in Iași, the son of Elena Bogdan (1907–2000), a chemistry professor at the University of Iași, and Sergiu-Andrei Culianu (1904–1964), a lawyer and a teacher. His maternal grandfather was Petru Bogdan, a chemistry professor and a Mayor of Iași, while one of his paternal grand-grandfathers was Nicolae Culianu, a professor of mathematics and astronomy. He studied at the University of Bucharest, graduating in May 1972 with a major in Italian language and literature. He then traveled to Italy, where he was granted political asylum while attending lectures in Perugia in July 1972. He graduated in November 1975 from the Università Cattolica del Sacro Cuore in Milan with a doctorate in the history of religion; his thesis, Gnosticismo e pensiero contemporaneo: Hans Jonas, was written under the direction of Ugo Bianchi. Culianu lived briefly in France and from 1976 to 1985 he taught at the University of Groningen in the Netherlands. He left Europe for the United States in 1986, becoming a permanent resident in January 1991. After a stint as visiting professor at the University of Chicago, he became a professor there; he was due to receive a permanent appointment in July 1991. He took a second PhD at the University of Paris IV in January 1987, with the thesis Recherches sur les dualismes d'Occident. Analyse de leurs principaux mythes ("Research into Western Dualisms. An Analysis of their Major Myths"), coordinated by Michel Meslin. Having completed three doctorates and being proficient in six languages, Culianu specialized in Renaissance magic and mysticism. He became a friend, and later the literary executor, of Mircea Eliade, the famous historian of religions. He also wrote fiction and political articles. Culianu had divorced his first wife, and at the time of his death was engaged to Hillary Wiesner, a 27-year-old graduate student at Harvard University. === Death === On Tuesday, May 21, 1991, at noon, just minutes after concluding a conversation with his doctoral student, Alexander Argüelles, on a day when the building was teeming with visitors to a book sale, Culianu was murdered in the bathroom of Swift Hall, of the University of Chicago Divinity School. He was shot once in the back of the head with a .25 caliber automatic weapon. The identity of the killer and the motive are still unknown. Speculation arose that he had been killed by former Securitate agents, due to political articles in which he attacked the Communist regime. The murder occurred a year and a half after the Romanian Revolution of December 1989 and Nicolae Ceaușescu's death. Before being killed, he had published a number of articles and interviews that heavily criticized the Ion Iliescu post-Revolution regime, making Culianu one of the government's most vocal adversaries. Several theories link his murder with the Romanian Intelligence Service, which was widely perceived as the successor of the Securitate; several pages of Culianu's Securitate files are inexplicably missing. Some reports suggest that Culianu had been threatened by anonymous phone calls in the days leading up to his killing. Ultra-nationalist and neo-fascist involvement, as part of an Iron Guard revival in connection with the nationalist discourse of the late years of Ceauşescu's rule and the rise of the Vatra Românească and România Mare parties, was not itself excluded from the scenario; according to Vladimir Tismăneanu: "[Culianu] gave the most devastating indictment of the new union of far left and far right in Romania". As part of his criticism of the Iron Guard, Culianu had come to expose Mircea Eliade's connections with the latter movement during the interwar years (because of this, relations between the two academics had soured for the final years of Eliade's life). Culianu was buried at Eternitatea Cemetery in Iași. == Works == === Scholarly works === Mircea Eliade, Assisi, Cittadella Editrice, 1978; Roma, Settimo Sigillo, 2008(2); Mircea Eliade, București, Nemira, 1995, 1998(2); Iași, Polirom, 2004(3) Iter in Silvis: Saggi scelti sulla gnosi e altri studi, Gnosis, no. 2, Messina, EDAS, 1981 Religione e accrescimento del potere, in G. Romanato, M. Lombardo, I.P. Culianu, Religione e potere, Torino, Marietti, 1981 Psychanodia: A Survey of the Evidence Concerning the Ascension of the Soul and Its Relevance, Leiden, Brill, 1983 Éros et Magie à la Renaissance. 1484, Paris, Flammarion, 1984; Eros and Magic in the Renaissance, Chicago, University of Chicago Press, 1987; Eros e magia nel Rinascimento: La congiunzione astrologica del 1484, Milano, Il Saggiatore – A. Mondadori, 1987; Eros şi magie în Renaştere. 1484, București, Nemira, 1994, 1999(2); Iaşi, Polirom, 2003(3), 2011, 2015; Eros y magia en el Renacimiento. 1484, Madrid, Ediciones Siruela, 1999 Expériences de l'extase: Extase, ascension et récit visionnaire de l'hellénisme au Moyen Age, Paris, Payot, 1984; Experienze dell'estasi dall'Ellenismo al Medioevo, Bari, Laterza, 1986 Gnosticismo e pensiero moderno: Hans Jonas, Roma, L'Erma di Bretschneider, 1985 Recherches sur les dualismes d'Occident: Analyse de leurs principaux mythes, Lille, Lille-Thèses, 1986; I miti dei dualismi occidentali, Milano, Jaca Book, 1989 Les Gnoses dualistes d'Occident: Histoire et mythes, Paris, Plon, 1990; The Tree of Gnosis, New York, HarperCollins, 1992; Gnozele dualiste ale Occidentului. Istorie si mituri, București, Nemira, 1995; Iaşi, Polirom, 2002(2), 2013 Out of this World: Otherworldly Journeys from Gilgamesh to Albert Einstein, Boston, Shambhala, 1991; Mas alla de este mundo, Barcelona, Paidos Orientalia, 1993; Călătorii in lumea de dincolo, București, Nemira, 1994, 1999(2); Iași, Polirom, 2003(3), 2007, 2015; Jenseits dieser Welt, Munchen, Eugen Diederichs Verlag, 1995 I viaggi dell'anima, Milano, Arnoldo Mondadori Editore, 1991 The Tree of Gnosis : Gnostic Mythology from Early Christianity to Modern Nihilism, San Francisco, HarperCollins, 1992; Arborele gnozei. Mitologia gnostică de la creştinismul timpuriu la nihilismul modern, București, Nemira, 1999; Iasi, Polirom, 2005, 2015 Experiences del extasis, Barcelona, Paidos Orientalia, 1994; Experienţe ale extazului, București, Nemira, 1997; Iași, Polirom, 2004(2) Religie şi putere, București, Nemira, 1996; Iași, Polirom, 2005 Psihanodia, București, Nemira, 1997, Iași, Polirom, 2006 Păcatul împotriva spiritului. Scrieri politice, București, Nemira, 1999; Iași, Polirom, 2005, 2013 Studii româneşti I. Fantasmele nihilismului. Secretul Doctorului Eliade, București, Nemira, 2000; Iași, Polirom, 2006 "Studii românești II. Soarele și Luna. Otrăvurile admirației", Iași, Polirom, 2009 "Iter in silvis I. Eseuri despre gnoză și alte studii", Iași, Polirom, 2012 "Iter in silvis II. Gnoză și magie", Iași, Polirom, 2013 Jocurile minţii. Istoria ideilor, teoria culturii, epistemologie, Iaşi, Polirom, 2002 Iocari serio. Ştiinţa şi arta în gîndirea Renaşterii, Iaşi, Polirom, 2003 Cult, magie, erezii. Articole din enciclopedii ale religiilor, Iaşi, Polirom, 2003 ``Marsilio Ficino (1433–1499) si problemele platonismului in Renastere, Iasi, Polirom, 2015 ``Dialoguri intrerupte. Corespondenta Mircea Eliade-Ioan Petru Culianu, Iasi, Polirom, 2004, 2013 (2) ==== Co-author ==== With Mircea Eliade and H.S. Wiesner: Dictionnaire des Religions, Avec la collaboration de H.S. Wiesner. Paris, Plon, 1990, 1992(2); The Eliade Guide to World Religions, Harper, San Francisco, 1991; Handbuch der Religionen, Zürich und München, Artemis-Winkler-Verlag, 1991; Suhrkamp-Taschenbuch, 1995; Diccionario de las religiones Barcelona, Paidos Orientalia, 1993; Dicţionarul religiilor, București, Humanitas, 1993, 1996(2); Iași, Polirom, 2007 The Encyclopedia of Religion, Collier Macmillan, New York, 1987 The HarperCollins Concise Guide to World Religions, Harper, San Francisco, 2000 === Fiction === La collezione di smeraldi. Racconti, Milano, Jaca Book, 1989 Hesperus, București, Univers, 1992; București, Nemira, 1998(2); Iaşi, Polirom, 2003(3) Pergamentul diafan. Povestiri, București, Nemira, 1994 Pergamentul diafan. Ultimele povestiri, București, Nemira, 1996(2); Iaşi, Polirom, 2002(3) Arta fugii. Povestiri, Iași, Polirom, 2002 Jocul de smarald, Iași, Polirom, 2005, 2011(2) "Tozgrec, Iași, Polirom, 2010 === Other === Dialoguri întrerupte. Corespondența Mircea Eliade – Ioan Petru Culianu, Iași, Polirom, 2004 == Works about Culianu == A biography and an analysis of his death was published by Ted Anton under the title Eros, Magic, and the Murder of Professor Culianu, 2013, (alluding to Culianu's most influential work, Eros and Magic in the Renaissance). Saul Bellow alludes to Culianu's murder in his novella Ravelstein, 2000. Elemire Zolla, Ioan Petru Culianu, Alberto Tallone Editore, 1994. Umberto Eco, Murder in Chicago, in The New York Review of Books, April 10, 1997 Sorin Antohi (ed.), Religion, Fiction, and History. Essays in Memory of Ioan Petru Culianu, Volumes I-II, Bucharest, Nemira, 2001. Sorin Antohi (coordinator), Ioan Petru Culianu. Omul și opera, Iași, Polirom, 2003 Matei Călinescu, Despre Ioan Petru Culianu și Mircea Eliade. Amintiri, lecturi, reflecții, Iași, Polirom, 2002, 2005(2). Andrei Oișteanu, Religie, politică şi mit. Texte despre Mircea Eliade și Ioan Petru Culianu, Polirom, Iași, 2007 (ediția a doua, revăzuta, adăugita și ilustrată, Polirom, Iași, 2014) Marcello De Martino, Mircea Eliade esoterico. Ioan Petru Culianu e i "non detti", Roma, Settimo Sigillo, 2008. Olga Gorshunova, Terra Incognita of Ioan Culianu, in Etnograficheskoe Obozrenie, 2008 № 6:94-110. ISSN 0869-5415 Raul Popescu, Ioan Petru Culianu. Ipostazele unui eretic, Eikon, Bucuresti, 2017. == See also == List of homicides in Illinois == Notes == == References == Anton, Ted (1992). "The Killing of Professor Culianu". Lingua Franca. Vol. 2, no. 6. Anton, Ted (1996). Eros, Magic, and the Murder of Professor Culianu. Northwestern University Press. Antohi, Sorin (Spring 2001), Exploring the Legacy of Ioan Petru Culianu, Institut für die Wissenschaften vom Menschen Post, archived from the original on 2021-05-15, retrieved 2006-11-25 (in Russian) Olga Gorshunova, Ioan Culianu: v chetvjortom izmerenii Archived 2020-02-15 at the Wayback Machine (Ioan Culianu: In The Fourth Dimension), in Etnoragraficheskoe Obozrenie Online, September, 2008. Eco, Umberto (April 10, 1997), "Murder in Chicago" (review of Ted Anton's Eros, Magic, and the Murder of Professor Culianu)", The New York Review of Books, vol. 44, no. 6 == External links == David Levi Strauss, "Magic & Images/Images & Magic", in The Brooklyn Rail Guide to the Ioan P. Culianu Papers 1883-1991 at the University of Chicago Special Collections Research Center
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Wikipedia:Nicolae Popescu#0
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Nicolae Popescu (Romanian: [nikoˈla.e poˈpesku]; 22 September 1937 – 29 July 2010) was a Romanian mathematician and professor at the University of Bucharest. He also held a research position at the Institute of Mathematics of the Romanian Academy, and was elected corresponding Member of the Romanian Academy in 1997. He is best known for his contributions to algebra and the theory of abelian categories. From 1964 to 2007 he collaborated with Pierre Gabriel on the characterization of abelian categories; their best-known result is the Gabriel–Popescu theorem, published in 1964. His areas of expertise were category theory, abelian categories with applications to rings and modules, adjoint functors, limits and colimits, the theory of sheaves, the theory of rings, fields and polynomials, and valuation theory. He also had interests and published in algebraic topology, algebraic geometry, commutative algebra, K-theory, class field theory, and algebraic function theory. == Biography == Popescu was born on September 22, 1937, in Strehaia-Comanda, Mehedinți County, Romania. In 1954 he graduated from the Carol I High School in Craiova and went on to study mathematics at the University of Iași. In his third year of studies he was expelled from the university, having been deemed "hostile to the regime" for remarking that "the achievements of American scientists are also worth of consideration." He then went back home to Strehaia, where he worked for a year in a collective farm, after which he was admitted in 1959 at the University of Bucharest, only to start anew as a freshman. Popescu earned his M.S. degree in mathematics in 1964, and his Ph.D. degree in mathematics in 1967, with thesis Krull–Remak–Schmidt Theorem and Theory of Decomposition written under the direction of Gheorghe Galbură. He was awarded a D. Phil. degree (Doctor Docent) in 1972, also by the University of Bucharest. While still a student, Popescu focused on category theory. He first approached the general theory, with its connections to homological algebra and algebraic topology, then shifted his focus on theory of Abelian categories, being one of the main promoters of this theory in Romania. He carried out mathematics studies at the Institute of Mathematics of the Romanian Academy in the Algebra research group, and also had international collaborations on three continents. He shared many moral, ethical, and religious values with Alexander Grothendieck, who visited the Faculty of Mathematics in Bucharest in 1968. Like Grothendieck, he had a long-standing interest in category theory and number theory, and supported promising young mathematicians in his fields of interest. He also promoted the early developments of category theory applications in relational biology and mathematical biophysics/mathematical biology. == Academic positions == Popescu was appointed as a Lecturer at the University of Bucharest in 1968 where he taught graduate students until 1972. Starting in 1964 he also held a research appointment at the Institute of Mathematics of the Romanian Academy. The institute was closed in 1976 by order of Nicolae Ceaușescu (for reasons related to his daughter Zoia Ceaușescu, who had been hired at the institute two years before), but was reopened in 1990, after the Romanian Revolution. == Publications == Between 1962 and 2008 Popescu published more than 102 papers in peer-reviewed mathematics journals, several monographs on the theory of sheaves, and several books on abelian category theory and abstract algebra, including Popescu, Nicolae (1968). Elemente de teoria algebrică a numerelor [Elements of the Algebraic Theory of Numbers] (in Romanian). Universitatea București. Popescu, Nicolae; Radu, Alexandru (1971). Teoria categoriilor și a fasciculelor [Category Theory and Sheaves] (in Romanian). Editura Științifică. Popescu, Nicolae (1971). Categorii abeliene [Abelian Categories] (in Romanian). Editura Academiei. MR 0322011. Popescu, Nicolae (1973). Abelian categories with applications to rings and modules. London Mathematical Society Monographs. Vol. 3. London: Academic Press. ISBN 0-12-561550-7. MR 0340375. Popescu, Nicolae; Popescu, Liliana (1979). Theory of categories. Martinus Nijhoff Publishers, The Hague; Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn. doi:10.1007/978-94-009-9550-5. ISBN 90-286-0168-6. MR 0538276. In a Grothendieck-like, energetic style, he initiated and provided scientific leadership to several seminars on category theory, sheaves and abstract algebra which resulted in a continuous stream of high-quality mathematical publications in international, peer-reviewed mathematics journals by several members participating in his Seminar series. == Personal life == Popescu died in Bucharest on July 29, 2010. He is survived by his wife, Professor Dr. Elena Liliana Popescu (a mathematician, poet, literary translator and editor), and their three children, one of whom, Dan Cristian Popescu, is a politician. == Recognition == In 1971 Popescu received the Simion Stoilow Prize in Mathematics of the Romanian Academy. He was elected President of the Romanian Mathematical Society in 1990 and corresponding Member of the Romanian Academy in 1997. On the 80th anniversary of his birthday, the Faculty of Mathematics and Informatics at the University of Bucharest and the Institute of Mathematics of the Romanian Academy organized a conference in his memory. == Notes ==
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Wikipedia:Nicolas Bourbaki#0
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Nicolas Bourbaki (French: [nikola buʁbaki]) is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics. The series is known collectively as the Éléments de mathématique (Elements of Mathematics), the group's central work. Topics treated in the series include set theory, abstract algebra, topology, analysis, Lie groups and Lie algebras. Bourbaki was founded in response to the effects of the First World War which caused the death of a generation of French mathematicians; as a result, young university instructors were forced to use dated texts. While teaching at the University of Strasbourg, Henri Cartan complained to his colleague André Weil of the inadequacy of available course material, which prompted Weil to propose a meeting with others in Paris to collectively write a modern analysis textbook. The group's core founders were Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné and Weil; others participated briefly during the group's early years, and membership has changed gradually over time. Although former members openly discuss their past involvement with the group, Bourbaki has a custom of keeping its current membership secret. The group's name derives from the 19th century French general Charles-Denis Bourbaki, who had a career of successful military campaigns before suffering a dramatic loss in the Franco-Prussian War. The name was therefore familiar to early 20th-century French students. Weil remembered an ENS student prank in which an upperclassman posed as a professor and presented a "theorem of Bourbaki"; the name was later adopted. The Bourbaki group holds regular private conferences for the purpose of drafting and expanding the Éléments. Topics are assigned to subcommittees, drafts are debated, and unanimous agreement is required before a text is deemed fit for publication. Although slow and labor-intensive, the process results in a work which meets the group's standards for rigour and generality. The group is also associated with the Séminaire Bourbaki, a regular series of lectures presented by members and non-members of the group, also published and disseminated as written documents. Bourbaki maintains an office at the ENS. Nicolas Bourbaki was influential in 20th-century mathematics, particularly during the middle of the century when volumes of the Éléments appeared frequently. The group is noted among mathematicians for its rigorous presentation and for introducing the notion of a mathematical structure, an idea related to the broader, interdisciplinary concept of structuralism. Bourbaki's work informed the New Math, a trend in elementary math education during the 1960s. Although the group remains active, its influence is considered to have declined due to infrequent publication of new volumes of the Éléments. However, since 2012 the group has published four new (or significantly revised) volumes, the most recent in 2023 (treating spectral theory). Moreover, at least three further volumes are under preparation. == Background == Charles-Denis Sauter Bourbaki was a successful general during the era of Napoleon III, serving in the Crimean War and other conflicts. During the Franco-Prussian war however, Charles-Denis Bourbaki suffered a major defeat in which the Armée de l'Est, under his command, retreated across the Swiss border and was disarmed. The general unsuccessfully attempted suicide. The dramatic story of his defeat was known in the French popular consciousness following his death. In the early 20th century, the First World War affected Europeans of all professions and social classes, including mathematicians and male students who fought and died in the front. For example, the French mathematician Gaston Julia, a pioneer in the study of fractals, lost his nose during the war and wore a leather strap over the affected part of his face for the rest of his life. The deaths of ENS students resulted in a lost generation in the French mathematical community; the estimated proportion of ENS mathematics students (and French students generally) who died in the war ranges from one-quarter to one-half, depending on the intervals of time (c. 1900–1918, especially 1910–1916) and populations considered. Furthermore, Bourbaki founder André Weil remarked in his memoir Apprenticeship of a Mathematician that France and Germany took different approaches with their intelligentsia during the war: while Germany protected its young students and scientists, France instead committed them to the front, owing to the French culture of egalitarianism. A succeeding generation of mathematics students attended the ENS during the 1920s, including Weil and others, the future founders of Bourbaki. During his time as a student, Weil recalled a prank in which an upperclassman, Raoul Husson, posed as a professor and gave a math lecture, ending with a prompt: "Theorem of Bourbaki: you are to prove the following...". Weil was also aware of a similar stunt around 1910 in which a student claimed to be from the fictional, impoverished nation of "Poldevia" and solicited the public for donations. Weil had strong interests in languages and Indian culture, having learned Sanskrit and read the Bhagavad Gita. After graduating from the ENS and obtaining his doctorate, Weil took a teaching stint at the Aligarh Muslim University in India. While there, Weil met the mathematician Damodar Kosambi, who was engaged in a power struggle with one of his colleagues. Weil suggested that Kosambi write an article with material attributed to one "Bourbaki", in order to show off his knowledge to the colleague. Kosambi took the suggestion, attributing the material discussed in the article to "the little-known Russian mathematician D. Bourbaki, who was poisoned during the Revolution." It was the first article in the mathematical literature with material attributed to the eponymous "Bourbaki". Weil's stay in India was short-lived; he attempted to revamp the mathematics department at Aligarh, without success. The university administration planned to fire Weil and promote his colleague Vijayaraghavan to the vacated position. However, Weil and Vijayaraghavan respected one another. Rather than play any role in the drama, Vijayaraghavan instead resigned, later informing Weil of the plan. Weil returned to Europe to seek another teaching position. He ended up at the University of Strasbourg, joining his friend and colleague Henri Cartan. == The Bourbaki collective == === Founding === During their time together at Strasbourg, Weil and Cartan regularly complained to each other regarding the inadequacy of available course material for calculus instruction. In his memoir Apprenticeship, Weil described his solution in the following terms: "One winter day toward the end of 1934, I came upon a great idea that would put an end to these ceaseless interrogations by my comrade. 'We are five or six friends', I told him some time later, 'who are in charge of the same mathematics curriculum at various universities. Let us all come together and regulate these matters once and for all, and after this, I shall be delivered of these questions.' I was unaware of the fact that Bourbaki was born at that instant." Cartan confirmed the account. The first, unofficial meeting of the Bourbaki collective took place at noon on Monday, 10 December 1934, at the Café Grill-Room A. Capoulade, Paris, in the Latin Quarter. Six mathematicians were present: Henri Cartan, Claude Chevalley, Jean Delsarte, Jean Dieudonné, René de Possel, and André Weil. Most of the group were based outside Paris and were in town to attend the Julia Seminar, a conference prepared with the help of Gaston Julia at which several future Bourbaki members and associates presented. The group resolved to collectively write a treatise on analysis, for the purpose of standardizing calculus instruction in French universities. The project was especially meant to supersede the text of Édouard Goursat, which the group found to be badly outdated, and to improve its treatment of Stokes' Theorem. The founders were also motivated by a desire to incorporate ideas from the Göttingen school, particularly from exponents Hilbert, Noether and B.L. van der Waerden. Further, in the aftermath of World War I, there was a certain nationalist impulse to save French mathematics from decline, especially in competition with Germany. As Dieudonné stated in an interview, "Without meaning to boast, I can say that it was Bourbaki that saved French mathematics from extinction." Jean Delsarte was particularly favorable to the collective aspect of the proposed project, observing that such a working style could insulate the group's work against potential later individual claims of copyright. As various topics were discussed, Delsarte also suggested that the work begin in the most abstract, axiomatic terms possible, treating all of mathematics prerequisite to analysis from scratch. The group agreed to the idea, and this foundational area of the proposed work was referred to as the "Abstract Packet" (Paquet Abstrait). Working titles were adopted: the group styled itself as the Committee for the Treatise on Analysis, and their proposed work was called the Treatise on Analysis (Traité d'analyse). In all, the collective held ten preliminary biweekly meetings at A. Capoulade before its first official, founding conference in July 1935. During this early period, Paul Dubreil, Jean Leray and Szolem Mandelbrojt joined and participated. Dubreil and Leray left the meetings before the following summer, and were respectively replaced by new participants Jean Coulomb and Charles Ehresmann. The group's official founding conference was held in Besse-en-Chandesse, from 10 to 17 July 1935. At the time of the official founding, the membership consisted of the six attendees at the first lunch of 10 December 1934, together with Coulomb, Ehresmann and Mandelbrojt. On 16 July, the members took a walk to alleviate the boredom of unproductive proceedings. During the malaise, some decided to skinny-dip in the nearby Lac Pavin, repeatedly yelling "Bourbaki!" At the close of the first official conference, the group renamed itself "Bourbaki", in reference to the general and prank as recalled by Weil and others. During 1935, the group also resolved to establish the mathematical personhood of their collective pseudonym by getting an article published under its name. A first name had to be decided; a full name was required for publication of any article. To this end, René de Possel's wife Eveline "baptized" the pseudonym with the first name of Nicolas, becoming Bourbaki's "godmother". This allowed for the publication of a second article with material attributed to Bourbaki, this time under "his" own name. Henri Cartan's father Élie Cartan, also a mathematician and supportive of the group, presented the article to the publishers, who accepted it. At the time of Bourbaki's founding, René de Possel and his wife Eveline were in the process of divorcing. Eveline remarried to André Weil in 1937, and de Possel left the Bourbaki collective some time later. This sequence of events has caused speculation that de Possel left the group because of the remarriage, however this suggestion has also been criticized as possibly historically inaccurate, since de Possel is supposed to have remained active in Bourbaki for years after André's marriage to Eveline. === World War II === Bourbaki's work slowed significantly during the Second World War, though the group survived and later flourished. Some members of Bourbaki were Jewish and therefore forced to flee from certain parts of Europe at certain times. Weil, who was Jewish, spent the summer of 1939 in Finland with his wife Eveline, as guests of Lars Ahlfors. Due to their travel near the border, the couple were suspected as Soviet spies by Finnish authorities near the onset of the Winter War, and André was later arrested. According to an anecdote, Weil was to have been executed but for the passing mention of his case to Rolf Nevanlinna, who asked that Weil's sentence be commuted. However, the accuracy of this detail is dubious. Weil reached the United States in 1941, later taking another teaching stint in São Paulo from 1945 to 1947 before settling at the University of Chicago from 1947 to 1958 and finally the Institute for Advanced Study in Princeton, where he spent the remainder of his career. Although Weil remained in touch with the Bourbaki collective and visited Europe and the group periodically following the war, his level of involvement with Bourbaki never returned to that at the time of founding. Second-generation Bourbaki member Laurent Schwartz was also Jewish and found pickup work as a math teacher in rural Vichy France. Moving from village to village, Schwartz planned his movements in order to evade capture by the Nazis. On one occasion Schwartz found himself trapped overnight in a certain village, as his expected transportation home was unavailable. There were two inns in town: a comfortable, well-appointed one, and a very poor one with no heating and bad beds. Schwartz's instinct told him to stay at the poor inn; overnight, the Nazis raided the good inn, leaving the poor inn unchecked. Meanwhile, Jean Delsarte, a Catholic, was mobilized in 1939 as the captain of an audio reconnaissance battery. He was forced to lead the unit's retreat from the northeastern part of France toward the south. While passing near the Swiss border, Delsarte overheard a soldier say "We are the army of Bourbaki"; the 19th-century general's retreat was known to the French. Delsarte had coincidentally led a retreat similar to that of the collective's namesake. === Postwar until the present === Following the war, Bourbaki had solidified the plan of its work and settled into a productive routine. Bourbaki regularly published volumes of the Éléments during the 1950s and 1960s, and enjoyed its greatest influence during this period. Over time the founding members gradually left the group, slowly being replaced with younger newcomers including Jean-Pierre Serre and Alexander Grothendieck. Serre, Grothendieck and Laurent Schwartz were awarded the Fields Medal during the postwar period, in 1954, 1966 and 1950 respectively. Later members Alain Connes and Jean-Christophe Yoccoz also received the Fields Medal, in 1982 and 1994 respectively. The later practice of accepting scientific awards contrasted with some of the founders' views. During the 1930s, Weil and Delsarte petitioned against a French national scientific "medal system" proposed by the Nobel physics laureate Jean Perrin. Weil and Delsarte felt that the institution of such a system would increase unconstructive pettiness and jealousy in the scientific community. Despite this, the Bourbaki group had previously successfully petitioned Perrin for a government grant to support its normal operations. Like the founders, Grothendieck was also averse to awards, albeit for pacifist reasons. Although Grothendieck was awarded the Fields Medal in 1966, he declined to attend the ceremony in Moscow, in protest of the Soviet government. In 1988, Grothendieck rejected the Crafoord Prize outright, citing no personal need to accept prize money, lack of recent relevant output, and general distrust of the scientific community. Born to Jewish anarchist parentage, Grothendieck survived the Holocaust and advanced rapidly in the French mathematical community, despite poor education during the war. Grothendieck's teachers included Bourbaki's founders, and so he joined the group. During Grothendieck's membership, Bourbaki reached an impasse concerning its foundational approach. Grothendieck advocated for a reformulation of the group's work using category theory as its theoretical basis, as opposed to set theory. The proposal was ultimately rejected in part because the group had already committed itself to a rigid track of sequential presentation, with multiple already-published volumes. Following this, Grothendieck left Bourbaki "in anger". Biographers of the collective have described Bourbaki's unwillingness to start over in terms of category theory as a missed opportunity. However, Bourbaki has in 2023 announced that a book on category theory is currently under preparation (see below the last paragraph of this section). During the founding period, the group chose the Parisian publisher Hermann to issue installments of the Éléments. Hermann was led by Enrique Freymann, a friend of the founders willing to publish the group's project, despite financial risk. During the 1970s, Bourbaki entered a protracted legal battle with Hermann over matters of copyright and royalty payment. Although the Bourbaki group won the suit and retained collective copyright of the Éléments, the dispute slowed the group's productivity. Former member Pierre Cartier described the lawsuit as a pyrrhic victory, saying: "As usual in legal battles, both parties lost and the lawyer got rich." Later editions of the Éléments were published by Masson, and modern editions are published by Springer. From the 1980s through the 2000s, Bourbaki published very infrequently, with the result that in 1998 Le Monde pronounced the collective "dead". However, in 2012 Bourbaki resumed the publication of the Éléments with a revised chapter 8 of algebra, the first 4 chapters of a new book on algebraic topology, and two volumes on spectral theory (the first of which is an expanded and revised version of the edition of 1967 while the latter consist of three new chapters). Moreover, the text of the two latest volumes announces that books on category theory and modular forms are currently under preparation (in addition to the latter part of the book on algebraic topology). == Working method == Bourbaki holds periodic conferences for the purpose of expanding the Éléments; these conferences are the central activity of the group's working life. Subcommittees are assigned to write drafts on specific material, and the drafts are later presented, vigorously debated, and re-drafted at the conferences. Unanimous agreement is required before any material is deemed acceptable for publication. A given piece of material may require six or more drafts over a period of several years, and some drafts are never developed into completed work. Bourbaki's writing process has therefore been described as "Sisyphean". Although the method is slow, it yields a final product which satisfies the group's standards for mathematical rigour, one of Bourbaki's main priorities in the treatise. Bourbaki's emphasis on rigour was a reaction to the style of Henri Poincaré, who stressed the importance of free-flowing mathematical intuition at the cost of thorough presentation. During the project's early years, Dieudonné served as the group's scribe, authoring several final drafts which were ultimately published. For this purpose, Dieudonné adopted an impersonal writing style which was not his own, but which was used to craft material acceptable to the entire group. Dieudonné reserved his personal style for his own work; like all members of Bourbaki, Dieudonné also published material under his own name, including the nine-volume Éléments d'analyse, a work explicitly focused on analysis and of a piece with Bourbaki's initial intentions. Most of the final drafts of Bourbaki's Éléments carefully avoided using illustrations, favoring a formal presentation based only in text and formulas. An exception to this was the treatment of Lie groups and Lie algebras (especially in chapters 4–6), which did make use of diagrams and illustrations. The inclusion of illustration in this part of the work was due to Armand Borel. Borel was minority-Swiss in a majority-French collective, and self-deprecated as "the Swiss peasant", explaining that visual learning was important to the Swiss national character. When asked about the dearth of illustration in the work, former member Pierre Cartier replied: The Bourbaki were Puritans, and Puritans are strongly opposed to pictorial representations of truths of their faith. The number of Protestants and Jews in the Bourbaki group was overwhelming. And you know that the French Protestants especially are very close to Jews in spirit. The conferences have historically been held at quiet rural areas. These locations contrast with the lively, sometimes heated debates which have occurred. Laurent Schwartz reported an episode in which Weil slapped Cartan on the head with a draft. The hotel's proprietor saw the incident and assumed that the group would split up, but according to Schwartz, "peace was restored within ten minutes." The historical, confrontational style of debate within Bourbaki has been partly attributed to Weil, who believed that new ideas have a better chance of being born in confrontation than in an orderly discussion. Schwartz related another illustrative incident: Dieudonné was adamant that topological vector spaces must appear in the work before integration, and whenever anyone suggested that the order be reversed, he would loudly threaten his resignation. This became an in-joke among the group; Roger Godement's wife Sonia attended a conference, aware of the idea, and asked for proof. As Sonia arrived at a meeting, a member suggested that integration must appear before topological vector spaces, which triggered Dieudonné's usual reaction. Despite the historical culture of heated argument, Bourbaki thrived during the middle of the twentieth century. Bourbaki's ability to sustain such a collective, critical approach has been described as "something unusual", surprising even its own members. In founder Henri Cartan's words, "That a final product can be obtained at all is a kind of miracle that none of us can explain." It has been suggested that the group survived because its members believed strongly in the importance of their collective project, despite personal differences. When the group overcame difficulties or developed an idea that they liked, they would sometimes say l'esprit a soufflé ("the spirit breathes"). Historian Liliane Beaulieu noted that the "spirit"—which might be an avatar, the group mentality in action, or Bourbaki "himself"—was part of an internal culture and mythology which the group used to form its identity and perform work. === Humor === Humor has been an important aspect of the group's culture, beginning with Weil's memories of the student pranks involving "Bourbaki" and "Poldevia". For example, in 1939 the group released a wedding announcement for the marriage of "Betti Bourbaki" (daughter of Nicolas) to one "H. Pétard" (H. "Firecrackers" or "Hector Pétard"), a "lion hunter". Hector Pétard was itself a pseudonym, but not one originally coined by the Bourbaki members. The Pétard moniker was originated by Ralph P. Boas, Frank Smithies and other Princeton mathematicians who were aware of the Bourbaki project; inspired by them, the Princeton mathematicians published an article on the "mathematics of lion hunting". After meeting Boas and Smithies, Weil composed the wedding announcement, which contained several mathematical puns. Bourbaki's internal newsletter La Tribu has sometimes been issued with humorous subtitles to describe a given conference, such as "The Extraordinary Congress of Old Fogies" (where anyone older than 30 was considered a fogy) or "The Congress of the Motorization of the Trotting Ass" (an expression used to describe the routine unfolding of a mathematical proof, or process). During the 1940s–1950s, the American Mathematical Society received applications for individual membership from Bourbaki. They were rebuffed by J.R. Kline who understood the entity to be a collective, inviting them to re-apply for institutional membership. In response, Bourbaki floated a rumor that Ralph Boas was not a real person, but a collective pseudonym of the editors of Mathematical Reviews with which Boas had been affiliated. The reason for targeting Boas was because he had known the group in its earlier days when they were less strict with secrecy, and he'd described them as a collective in an article for the Encyclopædia Britannica. In November 1968, a mock obituary of Nicolas Bourbaki was released during one of the seminars. The group developed some variants of the word "Bourbaki" for internal use. The noun "Bourbaki" might refer to the group proper or to an individual member, e.g. "André Weil was a Bourbaki." "Bourbakist" is sometimes used to refer to members but also denotes associates, supporters, and enthusiasts. To "bourbakize" meant to take a poor existing text and to improve it through an editing process. Bourbaki's culture of humor has been described as an important factor in the group's social cohesion and capacity to survive, smoothing over tensions of heated debate. As of 2025, a Twitter account registered to "Betty_Bourbaki" provides regular updates on the group's activity. == Works == Bourbaki's work includes a series of textbooks, a series of printed lecture notes, journal articles, and an internal newsletter. The textbook series Éléments de mathématique (Elements of mathematics) is the group's central work. The Séminaire Bourbaki is a lecture series held regularly under the group's auspices, and the talks given are also published as lecture notes. Journal articles have been published with authorship attributed to Bourbaki, and the group publishes an internal newsletter La Tribu (The Tribe) which is distributed to current and former members. === Éléments de mathématique === The content of the Éléments is divided into books—major topics of discussion, volumes—individual, physical books, and chapters, together with certain summaries of results, historical notes, and other details. The volumes of the Éléments have had a complex publication history. Material has been revised for new editions, published chronologically out of order of its intended logical sequence, grouped together and partitioned differently in later volumes, and translated into English. For example, the second book on Algebra was originally released in eight French volumes: the first in 1942 being chapter 1 alone, and the last in 1980 being chapter 10 alone. This presentation was later condensed into five volumes with chapters 1–3 in the first volume, chapters 4–7 in the second, and chapters 8–10 each remaining the third through fifth volumes of that portion of the work. The English edition of Bourbaki's Algebra consists of translations of the three volumes consisting of chapters 1–3, 4–7 and 8, with chapters 9 and 10 unavailable in English as of 2025. When Bourbaki's founders began working on the Éléments, they originally conceived of it as a "treatise on analysis", the proposed work having a working title of the same name (Traité d'analyse). The opening part was to comprehensively deal with the foundations of mathematics prior to analysis, and was referred to as the "Abstract Packet". Over time, the members developed this proposed "opening section" of the work to the point that it would instead run for several volumes and comprise a major part of the work, covering set theory, abstract algebra, and topology. Once the project's scope expanded far beyond its original purpose, the working title Traité d'analyse was dropped in favor of Éléments de mathématique. The unusual, singular "Mathematic" was meant to connote Bourbaki's belief in the unity of mathematics. The first six books of the Éléments, representing the first half of the work, are numbered sequentially and ordered logically, with a given statement being established only on the basis of earlier results. This first half of the work bore the subtitle Les structures fondamentales de l’analyse (Fundamental Structures of Analysis), covering established mathematics (algebra, analysis) in the group's style. The second half of the work consists of unnumbered books treating modern areas of research (Lie groups, commutative algebra), each presupposing the first half as a shared foundation but without dependence on each other. This second half of the work, consisting of newer research topics, does not have a corresponding subtitle. The volumes of the Éléments published by Hermann were indexed by chronology of publication and referred to as fascicules: installments in a large work. Some volumes did not consist of the normal definitions, proofs, and exercises in a math textbook, but contained only summaries of results for a given topic, stated without proof. These volumes were referred to as Fascicules de résultats, with the result that fascicule may refer to a volume of Hermann's edition, or to one of the "summary" sections of the work (e.g. Fascicules de résultats is translated as "Summary of Results" rather than "Installment of Results", referring to the content rather than a specific volume). The first volume of Bourbaki's Éléments to be published was the Summary of Results in the Theory of Sets, in 1939. Similarly one of the work's later books, Differential and Analytic Manifolds, consisted only of two volumes of summaries of results, with no chapters of content having been published. Later installments of the Éléments appeared infrequently during the 1980s and 1990s. A volume of Commutative Algebra (chapters 8–9) was published in 1983, and no other volumes were issued until the appearance of the same book's tenth chapter in 1998. During the 2010s, Bourbaki increased its productivity. A re-written and expanded version of the eighth chapter of Algebra appeared in 2012, the first four chapters of a new book treating Algebraic Topology was published in 2016, and the first two chapters of a revised and expanded edition of Spectral Theory was issued in 2019 while the remaining three (completely new) chapters appeared in 2023. === Séminaire Bourbaki === The Séminaire Bourbaki has been held regularly since 1948, and lectures are presented by non-members and members of the collective. As of 2025 the Séminaire Bourbaki has run to over a thousand recorded lectures in its written incarnation, denoted chronologically by simple numbers. At the time of a June 1999 lecture given by Jean-Pierre Serre on the topic of Lie groups, the total lectures given in the series numbered 864, corresponding to roughly 10,000 pages of printed material. === Articles === Several journal articles have appeared in the mathematical literature with material or authorship attributed to Bourbaki; unlike the Éléments, they were typically written by individual members and not crafted through the usual process of group consensus. Despite this, Jean Dieudonné's essay "The Architecture of Mathematics" has become known as Bourbaki's manifesto. Dieudonné addressed the issue of overspecialization in mathematics, to which he opposed the inherent unity of mathematic (as opposed to mathematics) and proposed mathematical structures as useful tools which can be applied to several subjects, showing their common features. To illustrate the idea, Dieudonné described three different systems in arithmetic and geometry and showed that all could be described as examples of a group, a specific kind of (algebraic) structure. Dieudonné described the axiomatic method as "the 'Taylor system' for mathematics" in the sense that it could be used to solve problems efficiently. Such a procedure would entail identifying relevant structures and applying established knowledge about the given structure to the specific problem at hand. Kosambi, D.D. (1931). "On a Generalization of the Second Theorem of Bourbaki". Bulletin of the Academy of Sciences of the United Provinces of Agra and Oudh, Allahabad, India. 1: 145–47. ISBN 978-81-322-3674-0. {{cite journal}}: ISBN / Date incompatibility (help) Reprinted in Ramaswamy, Ramakrishna, ed. (2016). D.D. Kosambi: Selected Works in Mathematics and Statistics. Springer. pp. 55–57. doi:10.1007/978-81-322-3676-4_6. Kosambi attributed material in the article to "D. Bourbaki", the first mention of the eponymous Bourbaki in the literature. Bourbaki, Nicolas (1935). "Sur un théorème de Carathéodory et la mesure dans les espaces topologiques". Comptes rendus de l'Académie des Sciences. 201: 1309–11. Presumptive author: André Weil. —— (1938). "Sur les espaces de Banach". Comptes rendus de l'Académie des Sciences. 206: 1701–04. Presumptive author: Jean Dieudonné. ——; Dieudonné, Jean (1939). "Note de tératopologie II". Revue scientifique (Or, "Revue rose"): 180–81. Presumptive author: Jean Dieudonné. Second in a series of three articles. —— (1941). "Espaces minimaux et espaces complètement séparés". Comptes rendus de l'Académie des Sciences. 212: 215–18. Presumptive author: Jean Dieudonné or André Weil. —— (1948). "L'architecture des mathématiques". In Le Lionnais, François (ed.). Les grands courants de la pensée mathématique. Actes Sud. pp. 35–47. Presumptive author: Jean Dieudonné. —— (1949). "Foundations of Mathematics for the Working Mathematician". Journal of Symbolic Logic. 14 (1): 1–8. doi:10.2307/2268971. JSTOR 2268971. S2CID 26516355. Presumptive author: André Weil. —— (1949). "Sur le théorème de Zorn". Archiv der Mathematik. 2 (6): 433–37. doi:10.1007/BF02036949. S2CID 117826806. Presumptive author: Henri Cartan or Jean Dieudonné. —— (1950). "The Architecture of Mathematics". American Mathematical Monthly. 57 (4): 221–32. doi:10.1080/00029890.1950.11999523. JSTOR 2305937. Presumptive author: Jean Dieudonné. Authorized translation of the book chapter L'architecture des mathématiques, appearing in English as a journal article. —— (1950). "Sur certains espaces vectoriels topologiques". Annales de l'Institut Fourier. 2: 5–16. doi:10.5802/aif.16. Presumptive authors: Jean Dieudonné and Laurent Schwartz. === La Tribu === La Tribu is Bourbaki's internal newsletter, distributed to current and former members. The newsletter usually documents recent conferences and activity in a humorous, informal way, sometimes including poetry. Member Pierre Samuel wrote the newsletter's narrative sections for several years. Early editions of La Tribu and related documents have been made publicly available by Bourbaki. Historian Liliane Beaulieu examined La Tribu and Bourbaki's other writings, describing the group's humor and private language as an "art of memory" which is specific to the group and its chosen methods of operation. Because of the group's secrecy and informal organization, individual memories are sometimes recorded in a fragmentary way, and may not have significance to other members. On the other hand, the predominantly French, ENS background of the members, together with stories of the group's early period and successes, create a shared culture and mythology which is drawn upon for group identity. La Tribu usually lists the members present at a conference, together with any visitors, family members or other friends in attendance. Humorous descriptions of location or local "props" (cars, bicycles, binoculars, etc.) can also serve as mnemonic devices. == Membership == As of 2000, Bourbaki has had "about forty" members. Historically the group has numbered about ten to twelve members at any given point, although it was briefly (and officially) limited to nine members at the time of founding. Bourbaki's membership has been described in terms of generations: Bourbaki was always a very small group of mathematicians, typically numbering about twelve people. Its first generation was that of the founding fathers, those who created the group in 1934: Weil, Cartan, Chevalley, Delsarte, de Possel, and Dieudonné. Others joined the group, and others left its ranks, so that some years later there were about twelve members, and that number remained roughly constant. Laurent Schwartz was the only mathematician to join Bourbaki during the war, so his is considered an intermediate generation. After the war, a number of members joined: Jean-Pierre Serre, Pierre Samuel, Jean-Louis Koszul, Jacques Dixmier, Roger Godement, and Sammy Eilenberg. These people constituted the second generation of Bourbaki. In the 1950s, the third generation of mathematicians joined Bourbaki. These people included Alexandre Grothendieck, François Bruhat, Serge Lang, the American mathematician John Tate, Pierre Cartier, and the Swiss mathematician Armand Borel. After the first three generations there were roughly twenty later members, not including current participants. Bourbaki has a custom of keeping its current membership secret, a practice meant to ensure that its output is presented as a collective, unified effort under the Bourbaki pseudonym, not attributable to any one author (e.g. for purposes of copyright or royalty payment). This secrecy is also intended to deter unwanted attention which could disrupt normal operations. However, former members freely discuss Bourbaki's internal practices upon departure. Prospective members are invited to conferences and styled as guinea pigs, a process meant to vet the newcomer's mathematical ability. In the event of agreement between the group and the prospect, the prospect eventually becomes a full member. The group is supposed to have an age limit: active members are expected to retire at (or about) 50 years of age. At a 1956 conference, Cartan read a letter from Weil which proposed a "gradual disappearance" of the founding members, forcing younger members to assume full responsibility for Bourbaki's operations. This rule is supposed to have resulted in a complete change of personnel by 1958. However, historian Liliane Beaulieu has been critical of the claim. She reported never having found written affirmation of the rule, and has indicated that there have been exceptions. The age limit is thought to express the founders' intent that the project should continue indefinitely, operated by people at their best mathematical ability—in the mathematical community, there is a widespread belief that mathematicians produce their best work while young. Among full members there is no official hierarchy; all operate as equals, having the ability to interrupt conference proceedings at any point, or to challenge any material presented. However, André Weil has been described as "first among equals" during the founding period, and was given some deference. On the other hand, the group has also poked fun at the idea that older members should be afforded greater respect. Bourbaki conferences have also been attended by members' family, friends, visiting mathematicians, and other non-members of the group. Bourbaki is not known ever to have had any female members. == Influence and criticism == Bourbaki was influential in 20th century mathematics and had some interdisciplinary impact on the humanities and the arts, although the extent of the latter influence is a matter of dispute. The group has been praised and criticized for its method of presentation, its working style, and its choice of mathematical topics. === Influence === Bourbaki introduced several mathematical notations which have remained in use. Weil took the letter Ø of the Norwegian alphabet and used it to denote the empty set, ∅. This notation first appeared in the Summary of Results on the Theory of Sets, and remains in use. The words injective, surjective and bijective were introduced to refer to functions which satisfy certain properties. Bourbaki used simple language for certain geometric objects, naming them pavés (paving stones) and boules (balls) as opposed to "parallelotopes" or "hyperspheroids". Similarly in its treatment of topological vector spaces, Bourbaki defined a barrel as a set which is convex, balanced, absorbing, and closed. The group were proud of this definition, believing that the shape of a wine barrel typified the mathematical object's properties. Bourbaki also employed a "dangerous bend" symbol ☡ in the margins of its text to indicate an especially difficult piece of material. Bourbaki enjoyed its greatest influence during the 1950s and 1960s, when installments of the Éléments were published frequently. Bourbaki had some interdisciplinary influence on other fields, including anthropology and psychology. This influence was in the context of structuralism, a school of thought in the humanities which stresses the relationships between objects over the objects themselves, pursued in various fields by other French intellectuals. In 1943, André Weil met the anthropologist Claude Lévi-Strauss in New York, where the two undertook a brief collaboration. At Lévi-Strauss' request, Weil wrote a brief appendix describing marriage rules for four classes of people within Aboriginal Australian society, using a mathematical model based on group theory. The result was published as an appendix in Lévi-Strauss' Elementary Structures of Kinship, a work examining family structures and the incest taboo in human cultures. In 1952, Jean Dieudonné and Jean Piaget participated in an interdisciplinary conference on mathematical and mental structures. Dieudonné described mathematical "mother structures" in terms of Bourbaki's project: composition, neighborhood, and order. Piaget then gave a talk on children's mental processes, and considered that the psychological concepts he had just described were very similar to the mathematical ones just described by Dieudonné. According to Piaget, the two were "impressed with each other". The psychoanalyst Jacques Lacan liked Bourbaki's collaborative working style and proposed a similar collective group in psychology, an idea which did not materialize. Bourbaki was also cited by post-structuralist philosophers. In their joint work Anti-Oedipus, Gilles Deleuze and Félix Guattari presented a criticism of capitalism. The authors cited Bourbaki's use of the axiomatic method (with the purpose of establishing truth) as a distinct counter-example to management processes which instead seek economic efficiency. The authors said of Bourbaki's axiomatics that "they do not form a Taylor system", inverting the phrase used by Dieudonné in "The Architecture of Mathematics". In The Postmodern Condition, Jean-François Lyotard criticized the "legitimation of knowledge", the process by which statements become accepted as valid. As an example, Lyotard cited Bourbaki as a group which produces knowledge within a given system of rules. Lyotard contrasted Bourbaki's hierarchical, "structuralist" mathematics with the catastrophe theory of René Thom and the fractals of Benoit Mandelbrot, expressing preference for the latter "postmodern science" which problematized mathematics with "fracta, catastrophes, and pragmatic paradoxes". Although biographer Amir Aczel stressed Bourbaki's influence on other disciplines during the mid-20th century, Maurice Mashaal moderated the claims of Bourbaki's influence in the following terms: While Bourbaki's structures were often mentioned in social science conferences and publications of the era, it seems that they didn't play a real role in the development of these disciplines. David Aubin, a science historian who analyzed Bourbaki's role in the structuralist movement in France, believes Bourbaki's role was that of a "cultural connector". According to Aubin, while Bourbaki didn't have any mission outside of mathematics, the group represented a sort of link between the various cultural movements of the time. Bourbaki provided a simple and relatively precise definition of concepts and structures, which philosophers and social scientists believed was fundamental within their disciplines and in bridges among different areas of knowledge. Despite the superficial nature of these links, the various schools of structuralist thinking, including Bourbaki, were able to support each other. So, it is not a coincidence that these schools suffered a simultaneous decline in the late 1960s. The impact of "structuralism" on mathematics itself was also criticized. The mathematical historian Leo Corry argued that Bourbaki's use of mathematical structures was unimportant within the Éléments, having been established in Theory of Sets and cited infrequently afterwards. Corry described the "structural" view of mathematics promoted by Bourbaki as an "image of knowledge"—a conception about a scientific discipline—as opposed to an item in the discipline's "body of knowledge", which refers to the actual scientific results in the discipline itself. Bourbaki also had some influence in the arts. The literary collective Oulipo was founded on 24 November 1960 under circumstances similar to Bourbaki's founding, with the members initially meeting in a restaurant. Although several members of Oulipo were mathematicians, the group's purpose was to create experimental literature by playing with language. Oulipo frequently employed mathematically-based constrained writing techniques, such as the S+7 method. Oulipo member Raymond Queneau attended a Bourbaki conference in 1962. In 2016, an anonymous group of economists collaboratively wrote a note alleging academic misconduct by the authors and editor of a paper published in the American Economic Review. The note was published under the name Nicolas Bearbaki in homage to Nicolas Bourbaki. In 2018, the American musical duo Twenty One Pilots released a concept album named Trench. The album's conceptual framework was the mythical city of "Dema" ruled by nine "bishops"; one of the bishops was named "Nico", short for Nicolas Bourbaki. Another of the bishops was named Andre, which may refer to André Weil. Following the album's release, there was a spike in internet searches for "Nicolas Bourbaki". === Praise === Bourbaki's work has been praised by some mathematicians. In a book review, Emil Artin described the Éléments in broad, positive terms: Our time is witnessing the creation of a monumental work: an exposition of the whole of present day mathematics. Moreover this exposition is done in such a way that the common bond between the various branches of mathematics become clearly visible, that the framework which supports the whole structure is not apt to become obsolete in a very short time, and that it can easily absorb new ideas. Among the volumes of the Éléments, Bourbaki's work on Lie Groups and Lie Algebras has been identified as "excellent", having become a standard reference on the topic. In particular, former member Armand Borel described the volume with chapters 4–6 as "one of the most successful books by Bourbaki". The success of this part of the work has been attributed to the fact that the books were composed while leading experts on the topic were Bourbaki members. Jean-Pierre Bourguignon expressed appreciation for the Séminaire Bourbaki, saying that he'd learned a large amount of material at its lectures, and referred to its printed lecture notes regularly. He also praised the Éléments for containing "some superb and very clever proofs". === Criticism === Bourbaki has also been criticized by several mathematicians—including its own former members—for a variety of reasons. Criticisms have included the choice of presentation of certain topics within the Éléments at the expense of others, dislike of the method of presentation for given topics, dislike of the group's working style, and a perceived elitist mentality around Bourbaki's project and its books, especially during the collective's most productive years in the 1950s and 1960s. Bourbaki's deliberations on the Éléments resulted in the inclusion of some topics, while others were not treated. When asked in a 1997 interview about topics left out of the Éléments, former member Pierre Cartier replied: There is essentially no analysis beyond the foundations: nothing about partial differential equations, nothing about probability. There is also nothing about combinatorics, nothing about algebraic topology, nothing about concrete geometry. And Bourbaki never seriously considered logic. Dieudonné himself was very vocal against logic. Anything connected with mathematical physics is totally absent from Bourbaki's text. Although Bourbaki had resolved to treat mathematics from its foundations, the group's eventual solution in terms of set theory was attended by several problems. Bourbaki's members were mathematicians as opposed to logicians, and therefore the collective had a limited interest in mathematical logic. As Bourbaki's members themselves said of the book on set theory, it was written "with pain and without pleasure, but we had to do it." Dieudonné personally remarked elsewhere that ninety-five percent of mathematicians "don't care a fig" for mathematical logic. In response, logician Adrian Mathias harshly criticized Bourbaki's foundational framework, noting that it did not take Gödel's results into account. Bourbaki also influenced the New Math, a failed reform in Western mathematics education at the elementary and secondary levels, which stressed abstraction over concrete examples. During the mid-20th century, reform in basic math education was spurred by a perceived need to create a mathematically literate workforce for the modern economy, and also to compete with the Soviet Union. In France, this led to the Lichnerowicz Commission of 1967, headed by André Lichnerowicz and including some (then-current and former) Bourbaki members. Although Bourbaki members had previously (and individually) reformed math instruction at the university level, they had less direct involvement with implementation of the New Math at the primary and secondary levels. New Math reforms resulted in instructional material which was incomprehensible to both students and teachers, failing to meet the cognitive needs of younger students. The attempted reform was harshly criticized by Dieudonné and also by brief founding Bourbaki participant Jean Leray. Apart from French mathematicians, the French reforms also met with harsh criticism from Soviet-born mathematician Vladimir Arnold, who argued that in his time as a student and teacher in Moscow, the teaching of mathematics was firmly rooted in analysis and geometry, and interweaved with problems from classical mechanics; hence, the French reforms cannot be a legitimate attempt to emulate Soviet scientific education. In 1997, while speaking to a conference on mathematical teaching in Paris, he commented on Bourbaki by stating: "genuine mathematicians do not gang up, but the weak need gangs in order to survive." and suggested that Bourbaki's bonding over "super-abstractness" was similar to groups of mathematicians in the 19th century who had bonded over anti-Semitism. Dieudonné later regretted that Bourbaki's success had contributed to a snobbery for pure mathematics in France, at the expense of applied mathematics. In an interview, he said: "It is possible to say that there was no serious applied mathematics in France for forty years after Poincaré. There was even a snobbery for pure math. When one noticed a talented student, one would tell him 'You should do pure math.' On the other hand, one would advise a mediocre student to do applied math while thinking, "It's all that he can do! ... The truth is actually the reverse. You can't do good work in applied math until you can do good work in pure math." Claude Chevalley confirmed an elitist culture within Bourbaki, describing it as "an absolute certainty of our superiority over other mathematicians." Alexander Grothendieck also confirmed an elitist mentality within Bourbaki. Some mathematicians, especially geometers and applied mathematicians, found Bourbaki's influence to be stifling. Benoit Mandelbrot's decision to emigrate to the United States in 1958 was motivated in part by a desire to escape Bourbaki's influence in France. Several related criticisms of the Éléments have concerned its target audience and the intent of its presentation. Volumes of the Éléments begin with a note to the reader which says that the series "takes up mathematics at the beginning, and gives complete proofs" and that "the method of exposition we have chosen is axiomatic and abstract, and normally proceeds from the general to the particular." Despite the opening language, Bourbaki's intended audience are not absolute beginners in mathematics, but rather undergraduates, graduate students, and professors who are familiar with mathematical concepts. Claude Chevalley said that the Éléments are "useless for a beginner", and Pierre Cartier clarified that "The misunderstanding was that it should be a textbook for everybody. That was the big disaster." The work is divided into two halves. While the first half—the Structures fondamentales de l’analyse—treats established subjects, the second half deals with modern research areas like commutative algebra and spectral theory. This divide in the work is related to a historical change in the intent of the treatise. The Éléments' content consists of theorems, proofs, exercises and related commentary, common material in math textbooks. Despite this presentation, the first half was not written as original research but rather as a reorganized presentation of established knowledge. In this sense, the Éléments' first half was more akin to an encyclopedia than a textbook series. As Cartier remarked, "The misunderstanding was that many people thought it should be taught the way it was written in the books. You can think of the first books of Bourbaki as an encyclopedia of mathematics... If you consider it as a textbook, it's a disaster." The strict, ordered presentation of material in the Éléments' first half was meant to form the basis for any further additions. However, developments in modern mathematical research have proven difficult to adapt in terms of Bourbaki's organizational scheme. This difficulty has been attributed to the fluid, dynamic nature of ongoing research which, being new, is not settled or fully understood. Bourbaki's style has been described as a particular scientific paradigm which has been superseded in a paradigm shift. For example, Ian Stewart cited Vaughan Jones' novel work in knot theory as an example of topology which was done without dependence on Bourbaki's system. Bourbaki's influence has declined over time; this decline has been partly attributed to the absence of certain modern topics—such as category theory—from the treatise. Although multiple criticisms have pointed to shortcomings in the collective's project, one has also pointed to its strength: Bourbaki was a "victim of its own success" in the sense that it accomplished what it set out to do, achieving its original goal of presenting a thorough treatise on modern mathematics. These factors prompted biographer Maurice Mashaal to conclude his treatment of Bourbaki in the following terms: Such an enterprise deserves admiration for its breadth, for its enthusiasm and selflessness, for its strongly collective character. Despite some mistakes, Bourbaki did add a little to 'the honor of the human spirit'. In an era when sports and money are such great idols of civilization, this is no small virtue. == See also == Bourbaki–Witt theorem Jacobson–Bourbaki theorem Secret society Other collective mathematical pseudonyms Arthur Besse Blanche Descartes John Rainwater G. W. Peck == Notes == == References == == Bibliography == Aczel, Amir D. (2006). The Artist and the Mathematician: the Story of Nicolas Bourbaki, the Genius Mathematician Who Never Existed. Thunder's Mouth Press. ISBN 978-1560259312. Aubin, David (1997). "The Withering Immortality of Nicolas Bourbaki: A Cultural Connector at the Confluence of Mathematics, Structuralism, and the Oulipo in France" (PDF). Science in Context. 10 (2). Cambridge University Press: 297–342. doi:10.1017/S0269889700002660. S2CID 170683589. Beaulieu, Liliane (1993). "A Parisian Café and Ten Proto-Bourbaki Meetings (1934–1935)". The Mathematical Intelligencer. 15 (1): 27–35. doi:10.1007/BF03025255. S2CID 189888171. Beaulieu, Liliane (1999). "Bourbaki's Art of Memory" (PDF). Osiris. 14: 219–51. doi:10.1086/649309. S2CID 143559711. Borel, Armand (March 1998). "Twenty-Five Years with Nicolas Bourbaki, (1949–1973)" (PDF). Notices of the American Mathematical Society. 45 (3): 373–80. Bourbaki, Nicolas (1950). "The Architecture of Mathematics". American Mathematical Monthly. 57 (4): 221–32. doi:10.1080/00029890.1950.11999523. JSTOR 2305937. Presumptive author: Jean Dieudonné. Authorized translation of the book chapter L'architecture des mathématiques, appearing in English as a journal article. Corry, Leo (2004). "Nicolas Bourbaki: Theory of Structures". Modern Algebra and the Rise of Mathematical Structures. Springer. pp. 289–338. ISBN 978-3764370022. Corry, Leo (2009). "Writing the Ultimate Mathematical Textbook: Nicolas Bourbaki's Éléments de mathématique". In Robson, Eleanor; Stedall, Jacqueline (eds.). The Oxford Handbook of the History of Mathematics. Oxford University Press. pp. 565–87. ISBN 978-0199213122. Guedj, Denis (1985). "Nicholas Bourbaki, Collective Mathematician : an Interview with Claude Chevalley" (PDF). Mathematical Intelligencer. 7 (2). Translated by Gray, Jeremy: 18–22. doi:10.1007/BF03024169. S2CID 123548747. Krömer, Ralf (2006). "La "machine de Grothendieck" se fonde-t-elle seulement sur des vocables métamathématiques? Bourbaki et les catégories au cours des années cinquante". Revue d'histoire des mathématiques. 12: 119–162. doi:10.24033/rhm.38. Zbl 1177.01034. Mashaal, Maurice (2006). Bourbaki: a Secret Society of Mathematicians. American Mathematical Society. ISBN 978-0821839676. Senechal, Marjorie (1998). "The Continuing Silence of Bourbaki: an Interview with Pierre Cartier, June 18, 1997". Mathematical Intelligencer. 20: 22–28. doi:10.1007/BF03024395. S2CID 124159858. == External links == Official Website of L'Association des Collaborateurs de Nicolas Bourbaki "Archives of the association" (in French). Retrieved 22 March 2025.
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Wikipedia:Nicolas Fizes#0
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Nicolas Fizes (27 October 1648 in Frontignan – 1718) was a French professor of mathematics and hydrography, who lived under the reign of Louis XIV. He is especially known as the librettist who wrote L'Opéra de Frontignan (1670), a play in Occitan, dealing with a slight love intrigue, and an idyllic poem on the fountain of Frontignan. == Career == Nicolas Fizes's parents were carpenters in the French Navy. He studied with the Jesuits, and became engineer to armies and a doctor of law. In 1682 he held the first professorship of mathematics and hydrography in Montpellier. From 1689, he headed a school of hydrography in Frontignan. In the hall of the Town Hall, he taught a few young sailors the concepts of mathematics and astrology. But this school was previously free, and Fizes asked for a salary of 150 pounds a year, which led to a conflict with the consuls of Frontignan. The school only survived 7 years, and closed its doors in 1696. == Bibliography == Lucien Albagnac, Contribution à l'Histoire de Frontignan (no ISBN) André Cablat, René Michel, Maurice Nougaret, Jean Valette, La Petite Encyclopédie de Frontignan la Peyrade (no ISBN) == See also == Occitan literature
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Wikipedia:Nicolas Rashevsky#0
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Nicolas Rashevsky (November 9, 1899 – January 16, 1972) was an American theoretical physicist who was one of the pioneers of mathematical biology, and is also considered the father of mathematical biophysics and theoretical biology. == Academic career == He studied theoretical physics at the St. Vladimir Imperial University of Kyiv. He left Ukraine after the October Revolution, emigrating first to Turkey, then to Poland, France, and finally to the US in 1924. In USA he worked at first for the Westinghouse Research Labs in Pittsburgh where he focused on the theoretical physics modeling of the cell division and the mathematics of cell fission. He was awarded a Rockefeller Fellowship in 1934 and went to the University of Chicago to take up the appointment of assistant professor in the department of physiology. In 1938, inspired by reading On Growth and Form (1917) by D'Arcy Wentworth Thompson, he made his first major contribution by publishing his first book on Mathematical Biophysics, and then in 1939 he also founded the first mathematical biology international journal entitled The Bulletin of Mathematical Biophysics (BMB); these two essential contributions founded the field of mathematical biology, with the Bulletin of Mathematical Biology serving as the focus of contributing mathematical biologists over the last 70 years. During the late 1930s, Rashevsky's research group was producing papers that had difficulty publishing in other journals at the time, so Rashevsky decided to found a new journal exclusively devoted to mathematical biophysics. In January 1939, he approached the editor of the journal Psychometrika, L.L. Thurstone, and formed an agreement that the new journal, the BMB, would be published as a supplement to their quarterly issues. === Major scientific contributions === In 1938 he published one of the first books on mathematical biology and mathematical biophysics entitled: "Mathematical Biophysics: Physico-Mathematical Foundations of Biology." This fundamental book was eventually published in three revised editions, the last revision appearing in two volumes in 1960. It was followed in 1940 by "Advances and applications of mathematical biology.", and in 1947 by "Mathematical theory of human relations", an approach to a mathematical model of society. In the same year he established the World' s first PhD program in mathematical biology at the University of Chicago. In the early 1930s, Rashevsky developed the first model of neural networks. This was paraphrased in a Boolean context by his student Walter Pitts together with Warren McCulloch, in A logical calculus of the ideas immanent in nervous activity, published in Rashevsky's Bulletin of Mathematical Biophysics in 1943. The Pitts-McCulloch article subsequently became extremely influential for research on artificial intelligence and artificial neural networks. His later efforts focused on the topology of biological systems, the formulation of fundamental principles in biology, relational biology, set theory and propositional logic formulation of the hierarchical organization of organisms and human societies. In the second half of the 1960s, he introduced the concept of "organismic sets" that provided a unified framework for physics, biology and sociology. This was subsequently developed by other authors as organismic supercategories and Complex Systems Biology. === Rashevsky's most notable students === Some of Rashevsky's most outstanding PhD students who earned their doctorate under his supervision were: George Karreman, Herbert Daniel Landahl, Clyde Coombs, Robert Rosen and Anatol Rapoport. In 1948, Anatol Rapoport took over Rashevsky's course in mathematical biology, so that Rashevsky could teach mathematical sociology instead. === Administrative and political obstacles === However, his more advanced ideas and abstract relational biology concepts found little support in the beginning amongst practicing experimental or molecular biologists, although current developments in complex systems biology clearly follow in his footsteps. In 1954 the budget for his Committee of Mathematical Biology was drastically cut; however, this was at least in part politically imposed, rather than scientifically, motivated. Thus, the subsequent University of Chicago administration—notably represented by the genetics Nobel laureate George Wells Beadle— who reversed in the 1960s the previous position and quadrupled the financial support for Rashevsky's Committee for Mathematical Biology research activities ("Reminiscences of Nicolas Rashevsky." by Robert Rosen, written in late 1972). There was later however a fall out between the retiring Nicolas Rashevsky and the University of Chicago president over the successor to the Chair of the Committee of Mathematical Biology; Nicolas Rashevsky strongly supported Dr. Herbert Landahl-his first PhD student to graduate in Mathematical Biophysics, whereas the president wished to appoint a certain US biostatistician. The result was Rashevsky's move to the University of Michigan in Ann Arbor, Michigan, and his taking ownership of the well-funded "Bulletin of Mathematical Biophysics". === Formation of Mathematical Biology, Inc. === He also formed in 1969 a non-profit organization, "Mathematical Biology, Incorporated", which was to be the precursor of "The Society for Mathematical Biology", with the purpose of "dissemination of information regarding Mathematical Biology". In his later years, after 1968, he became again very active in relational biology and held, as well as Chaired, in 1970 the first international "Symposium of Mathematical Biology" at Toledo, Ohio, in USA with the help of his former PhD student, Dr. Anthony Bartholomay, who has become the chairman of the first Department of Mathematical Medicine at Ohio University. The meeting was sponsored by Mathematical Biology, Inc. === Final quest for principles of biology === Rashevsky was greatly influenced and inspired both by Herbert Spencer's book on the Principles of Biology (1898), and also by J. H. Woodger `axiomatic (Mendelian) genetics', to launch his own search and quest for biological principles, and also to formulate mathematically precise principles and axioms of biology. He then developed his own highly original approach to address the fundamental question of What is Life? that another theoretical physicist, Erwin Schrödinger, had asked before him from the narrower viewpoint of quantum theory in biology. He wished to reach this `holy grail' of (theoretical/ mathematical) biology, but his heavy work load during the late 1960s—despite his related health problems—took its toll, and finally prevented him in 1972 from reaching his ultimate goal. Rashevsky's relational approach represents a radical departure from reductionistic approaches, and it has greatly influenced the work of his student Robert Rosen. == Biography == In 1917, Nicolas Rashevsky joined the White Russian Navy and in 1920 he and his wife, Countess Emily had to flee for their lives to Constantinople where he taught at the American College. In 1921 they moved to Prague where he taught both special and general relativity. From Prague, he moved in the 1930s to Paris, France, and then to New York, Pittsburgh and Chicago, USA. His life has been dedicated to the science that he founded, Mathematical Biology, and his wife Emily was very supportive and appreciative of his scientific efforts, accompanying him at the scientific meetings that he either initiated or attended. He cut a tall, impressive figure with a slight Eastern European accent, but a clear voice and thought to the very day when in 1972 he died from a heart attack caused by coronary heart disease. His generosity was very well known and is often recognized in print by former associates or visitors. As the Chief Editor of BMB he had a declared policy of helping the authors to optimize their presentation of submitted papers, as well as proving many valuable suggestions to the submitting authors. His suggested detailed changes, additions and further developments were like a real `gold mine' for the submitting authors. He managed to stay aloof of all science `politics' most of the time, even in very adverse circumstances such as those during the McCarthy era when completely unfounded political accusations were made about one or two members of his close research group. Not unlike another American theoretical physicist Robert Oppenheimer, he then had much to lose for his loyal support of the wrongly accused researcher in his group. == Works == Physico-mathematical aspects of Excitation and Conduction in Nerves., Cold Springs Harbor Symposia on Quantitative Biology.IV: Excitation Phenomena., 1936, p.90. Mathematical Biophysics:Physico-Mathematical Foundations of Biology. Univ. of Chicago Press. : Chicago Press, 1938/1948 (2nd ed.). Mathematical Theory of Human Relations: An Approach to Mathematical Biology of Social Phenomena. Bloomington, ID: Principia Press, 1947/1949 (2nd ed.) Topology and life: In search of general mathematical principles in biology and sociology. Bulletin of Mathematical Biophysics 16 (1954): 317–348. Proceedings of the International School of Physics "Enrico Fermi", Course 16, Physico-Mathematical Aspects of Biology. : Academic Press, 1964 Some Medical Aspects of Mathematical Biology. Springfield, IL: Charles C. Thomas, 1964 The Representation of Organisms in Terms of Predicates, Bulletin of Mathematical Biophysics 27 (1965): 477–491. Outline of a Unified Approach to Physics, Biology and Sociology., Bulletin of Mathematical Biophysics 31 (1969): 159–198. Looking at History through Mathematics, 1972 Organismic Sets., William Clowes & Sons., London, Beccles and Cochester, 1972. == Notes and references == This article incorporates material from Nicolas Rashevsky on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License. The article also incorporates additional data from planetphysics.org; furthermore, both external entries are original, contributed objects in the public domain. == Further reading == Abraham, Tara H. (2004). "Nicolas Rashevsky's Mathematical Biophysics". Journal of the History of Biology. 37 (2). Springer Science and Business Media LLC: 333–385. doi:10.1023/b:hist.0000038267.09413.0d. ISSN 0022-5010. Bartholomay, A. F., G. Karreman and H. D. Landahl (1972). "Obituary of Nicolas Rashevsky.", Bull. Math. Biophys. 34. Rosen, Robert. 1972. Tribute to Nicolas Rashevsky 1899–1972. Progress in Theoretical Biology 2. Tara H. Abraham. 2004. Journal of the History of Biology, 37: 333–385.[1] Rosen Robert. 1972. "Reminiscences of Nicolas Rashevsky", unpublished paper. Rosen, Robert. 1958. The representation of biological systems from the standpoint of the theory of categories. Bulletin of Mathematical Biophysics 20: 317–341. Natural Transformations of Organismic Structures., Bulletin of Mathematical Biology, 42: 431–446, Baianu, I.C.: 1980. Elsasser, M.W.: 1981, A Form of Logic Suited for Biology., In: Robert, Rosen, ed., Progress in Theoretical Biology, Volume 6, Academic Press, New York and London, pp 23–62. Rosen, Robert. 1985. The physics of complexity. Systems Research 2: 171–175. Rosen, Robert. 1985. Organisms as causal systems which are not mechanisms. In R. Rosen, Theoretical Biology and Complexity, 165–203. Rosen, Robert. 1979. Biology and system theory: An overview. In Klir, Proceedings of the System Theory Conference — Applied General Systems Research, Rosen, Robert. 1977. Complexity as a system property. International Journal of General Systems 3: 227–232. Rosen, Robert. 1977. Complexity and system description. In Hartnett, Systems, 169–175. Rosen, R. 1973. A unified approach to physics, biology, and sociology. In Rosen, Foundations of Mathematical Biology, 177–190. Rosen, R. 1972.Quantum genetics. In R. Rosen, Foundations of Mathematical Biology, 215–252. Rosen, R. 1972. Morphogenesis. In Rosen, Foundations of Mathematical Biology, 1–77. Rosen, R. 1972. Mechanics of epigenetic control. In R. Rosen, Foundations of Mathematical Biology, 79–140. == External links == Books by Rashevsky The Bulletin of Mathematical Biophysics Rashevsky's theory of two-factor systems for neural networks Guide to the Nicolas Rashevsky Papers 1920-1972 at the University of Chicago Special Collections Research Center
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Wikipedia:Nicolas de Malézieu#0
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Nicolas de Malézieu (or Malézieux) (or Malesieu) (7 September 1650, in Paris – 4 March 1727, in Paris) was a French intellectual, Greek scholar and mathematician. == Life and career == Nicolas de Malézieu was a squire and lord of Chatenay. He later became chancellor of Dombes and secretary-general to the Swiss and Grisons of France. He was the tutor of Louis Auguste, Duke of Maine (to whom he introduced Bossuet) and he declaimed the plays of Euripides and Sophocles to the duchess who had made her chateau of Sceaux into a literary salon. Here he became a member of the light-hearted fraternity she founded, the (fr) Knights of the Bee, and organised the festivals she loved, the fr:Grandes Nuits de Sceaux. Later tutor to duc de Bourgogne, he was appointed to the Académie royale des sciences in 1699 and to the Académie française in 1701. Malézieu collected and published the lessons in mathematics that he gave to the duc de Bourgogne over four years in 1705 as Élémens de géométrie de Mgr le duc de Bourgogne. Le Journal des savants reported in detail the observations he made in this work on geometry and infinitely small numbers. In 1713, this work was translated into Latin as Serenissimi Burgundiae Ducis Elementa Geometrica, ex Gallico Semone in Latinum translata ad Usum Seminarii Patavini. A third (posthumous) edition, with corrections and a supplemental treatise on logarithms, appeared 1729. Nicolas de Malézieu also translated Euripides’ Iphigenia in Tauris as well as poems, songs and sketches, which were published in 1712 in Les Divertissements de Sceaux and in 1725 in the Suite des Divertissements. Among these pieces are Philémon et Baucis, Le Prince de Cathay, Les Importuns de Chatenay, La Grande Nuit de l'éclipse, L'Hôte de Lemnos, La Tarentole and L'Heautontimorumenos. Often written in a single day, these pieces were set to music and staged for the amusement of the duchess, to whom Malézieu also gave courses in astronomy. A four-volume work, a Histoire des fermes du roi (History of Royal Farms) survives only a manuscript version dating from 1746. Pierre-Édouard Lémontey said of Malézieu "Knowing a bit about everything, he gathered in his servile person all the advantages of universal mediocrity." == Family == Malézieu was the son of Nicolas de Malézieu (1612-1652) and Marie des Forges (d.1680). His brother Michel Louis de Malézieu married Marie Jérônime Mac Carthy (d.1714) In 1672 Malézieu married Françoise Faudel de Fauveresse (1650-1741) by whom he had the following children: Nicolas de Malézieu (1674-1748), bishop of Lavaur en 1713. Pierre de Malézieu (1680-1756), married Louise Marthe Stoppa in 1717 (d.1720), under-secretary to the duc de Maine, secretary-general of the Swiss and Grisons in 1727, brigadier of the infantry, ifnanterie, maréchal de camp in 1734, lieutenant general of the artillery and commander of the Order of Saint-Louis in 1756. Charles-François de Malézieu (d.1763), lieutenant-colonel of a brigade of carabiniers, cavalry brigadier in 1745, governor of the harbour and defences of La Rochelle. Élisabeth de Malézieu (b.1676), married (1699) Antoine des Rioux de Missimy, first president of the parlement and intendant of Dombes Marie de Malézieu (b.1682), married (1705) Louis de Guiry, seigneur de Noncourt and la Roncière, mestre de camp of the cavalry, lieutenant general of Aunis and la Rochelle == External links == Les Divertissements de Sceaux, Trévoux, Étienne Ganeau, 1712 Members of the Académie des sciences == References ==
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Wikipedia:Nicolaus II Bernoulli#0
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Nicolaus II Bernoulli (also spelled as Niklaus or Nikolaus; 6 February 1695 in Basel – 31 July 1726 in Saint Petersburg) was a Swiss mathematician as were his father Johann Bernoulli and one of his brothers, Daniel Bernoulli. He was one of the many prominent mathematicians in the Bernoulli family. == Work == Nicolaus worked mostly on curves, differential equations, and probability. He was a friend and contemporary of Leonhard Euler, who studied under Nicolaus' father. He also contributed to fluid dynamics. He was older brother of Daniel Bernoulli, to whom he also taught mathematics. Even in his youth he had learned several languages. From the age of 13, he studied mathematics and law at the University of Basel. In 1711 he received his Master's of Philosophy; in 1715 he received a Doctorate in Law. In 1716-17 he was a private tutor in Venice. From 1719 he had the Chair in Mathematics at the University of Padua, as the successor of Giovanni Poleni. He served as an assistant to his father, among other areas, in the correspondence over the priority dispute between Isaac Newton and Leibniz, and also in the priority dispute between his father and the English mathematician Brook Taylor. In 1720 he posed the problem of reciprocal orthogonal trajectories, which was intended as a challenge for the English Newtonians. From 1723 he was a law professor at the Berner Oberen Schule. In 1725 he together with his brother Daniel, with whom he was touring Italy and France at this time, was invited by Peter the Great to the newly founded St. Petersburg Academy. Eight months after his appointment he came down with a fever and died. His professorship was succeeded in 1727 by Leonhard Euler, whom the Bernoulli brothers had recommended. His early death cut short a promising career. == See also == Bernoulli distribution Bernoulli process Bernoulli trial St. Petersburg paradox == External links == O'Connor, John J.; Robertson, Edmund F., "Nicolaus II Bernoulli", MacTutor History of Mathematics Archive, University of St Andrews Weisstein, Eric Wolfgang (ed.). "Bernoulli, Nicholas (1695-1726)". ScienceWorld. == Further reading == Fleckenstein, J.O. (1970–1980). "Bernoulli, Nickolaus II". Dictionary of Scientific Biography. Vol. 2. New York: Charles Scribner's Sons. pp. 57–58. ISBN 978-0-684-10114-9.
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Wikipedia:Nicole De Grande-De Kimpe#0
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Nicole Leonie Jean Marie De Grande-De Kimpe (7 September 1936 – 23 July 2008) was a Belgian mathematician known as a pioneer of p-adic functional analysis, and particularly for her work on locally convex topological vector spaces over fields with non-Archimedean valuations. == Early life and education == De Grande-De Kimpe was born on 7 September 1936 in Antwerp, the only child of a dockworker living in Hoboken, Antwerp, where she grew up, went to high school, and learned to play the violin. She studied mathematics on a scholarship to Ghent University, finished her degree with a specialty in mathematical analysis there in 1958, and took a job as a high school mathematics teacher. In 1963, she began a research fellowship, funded by the National Center for Algebra and Topology, which she used to study under Guy Hirsch at the Free University of Brussels. Following that, from 1965 to 1970 she worked as a graduate assistant in analysis for Piet Wuyts at the Free University of Brussels. During this time, she married, had a daughter, and divorced. She completed her Ph.D. in 1970, supervised by Hans Freudenthal at Utrecht University. == Later life and career == After a year of postdoctoral research with Freudenthal at Utrecht, De Grande-De Kimpe took a position in 1971 at the Vrije Universiteit Brussel, the Flemish half of the newly-split Free University of Brussels. There, she and Lucien Van Hamme organized a long-running seminar on p {\displaystyle p} -adic analysis beginning in 1978, and hosted an international conference on the subject in 1986. Recognizing that there was enough critical mass for a more specialized international conference on p {\displaystyle p} -adic functional analysis, De Grande-De Kimpe founded the series of International Conferences on p {\displaystyle p} -adic Functional Analysis, with cofounders Javier Martínez Maurica and José Manuel Bayod, beginning with the first conference in 1990 in Spain. She also served a term as head of the mathematics department at her university. After retiring to her home in Willebroek in 2001, she remained mathematically active and continued to teach the history of mathematics. She died on 23 July 2008. == Recognition == In 2002, a festschrift was published as a special volume of the Bulletin of the Belgian Mathematical Society, Simon Stevin, honoring both De Grande-De Kimpe and Lucien Van Hamme, who retired at the same time. The 11th International Conference on p {\displaystyle p} -adic Functional Analysis, held in 2010, was dedicated to the memory of De Grande-De Kimpe. == References ==
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Wikipedia:Nicole M. Joseph#0
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Nicole Michelle Joseph is an American mathematician and scholar of mathematics education whose research particularly focuses on the experiences of African-American girls and women in mathematics, on the effects of white supremacist reactions to their work in mathematics, and on the "intersectional nature of educational inequity". She is an associate professor of mathematics education, in the Department of Teaching and Learning of the Vanderbilt Peabody College of Education and Human Development. == Education and career == Joseph is African American, and is originally from Seattle. After a fall-out with a racist teacher in her elementary school, she was moved to the only open class, an advanced and self-paced classroom in which she first developed a love for mathematics. She majored in economics, with a minor in mathematics, at Seattle University, where she earned a bachelor's degree in business administration in 1993. After "a few years in the business world", she began working in the Seattle area as a middle school and elementary school mathematics teacher, and as a mathematics coach, from 1999 to 2011. During this period she also studied at Pacific Oaks College Northwest, a former Seattle satellite campus of Pacific Oaks College, a private Quaker college in California. Through Pacific Oaks, she earned a teaching certification for Washington in 2000, and a master's degree in human development in 2003. In 2011, Joseph completed a Ph.D. in Curriculum & Instruction at the University of Washington. Her dissertation, Black Students and Mathematics Achievement: A Mixed-Method Analysis of In-School and Out-of-School Factors Shaping Student Success, was supervised by James A. Banks. In the same year, she earned a national certification in adolescent mathematics teaching through the National Board for Professional Teaching Standards. After completing her doctorate, Joseph joined the University of Denver in 2011 as an assistant professor, focusing on educating future mathematics teachers. She moved to Vanderbilt University in 2016, and was tenured there as an associate professor in 2021. == Books == Joseph is the author or editor of books including: Interrogating Whiteness and Relinquishing Power: White Faculty's Commitment to Racial Consciousness in STEM Classrooms (edited with C. M. Haynes and F. Cobb, Peter Lang Publishers, 2016) Understanding the Intersections of Race, Gender, and Gifted Education: An Anthology by and About Talented Black Girls and Women in STEM (edited, Information Age Publishing, 2020) Making Black Girls Count in Math: A Black Feminist Vision of Transformative Teaching (Harvard Education Press, 2022) == Recognition == Joseph was the winner of the 2023 Louise Hay Award of the Association for Women in Mathematics, "recognized for her contributions to mathematics education that reflect the values of taking risks and nurturing students’ academic talent". == References == == External links == Home page Nicole M. Joseph publications indexed by Google Scholar
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Wikipedia:Nicole Tomczak-Jaegermann#0
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Nicole Tomczak-Jaegermann FRSC (8 June 1945 – 17 June 2022) was a Polish-Canadian mathematician, a professor of mathematics at the University of Alberta, and the holder of the Canada Research Chair in Geometric Analysis. == Contributions == Her research is in geometric functional analysis, and is unusual in combining asymptotic analysis with the theory of Banach spaces and infinite-dimensional convex bodies. It formed a key component of Fields medalist Timothy Gowers' solution to Stefan Banach's homogeneous space problem, posed in 1932. Her 1989 monograph on Banach–Mazur distances is also highly cited. == Education and career == Tomczak-Jaegermann earned her M.S. in 1968 from the University of Warsaw, and her Ph.D. from the same university in 1974, under the supervision of Aleksander Pełczyński. She remained on the faculty at the University of Warsaw from 1975 until 1983, when she moved to Alberta. == Recognition == In 1996, Tomczak-Jaegermann was elected to the Royal Society of Canada, and in 1999 she won the Krieger–Nelson Prize for an outstanding female Canadian mathematician. In 1998 she was an Invited Speaker of the International Congress of Mathematicians in Berlin. She was the winner of the 2006 CRM-Fields-PIMS prize for exceptional research in mathematics. == Death == Tomczak-Jaegermann died on 17 June 2022 at the age 77 in Edmonton, Alberta, Canada. == References == == External links == Home page at the University of Alberta Ghoussoub (8 August 2022). "Nicole Tomczak-Jaegermann 1945-2022". Piece of Mind.
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Wikipedia:Nicolo Tartaglia#0
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Nicolo, known as Tartaglia (Italian: [tarˈtaʎʎa]; 1499/1500 – 13 December 1557), was an Italian mathematician, engineer (designing fortifications), a surveyor (of topography, seeking the best means of defense or offense) and a bookkeeper from the then Republic of Venice. He published many books, including the first Italian translations of Archimedes and Euclid, and an acclaimed compilation of mathematics. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs, known as ballistics, in his Nova Scientia (A New Science, 1537); his work was later partially validated and partially superseded by Galileo's studies on falling bodies. He also published a treatise on retrieving sunken ships. == Personal life == Nicolo was born in Brescia, the son of Michele, a dispatch rider who travelled to neighbouring towns to deliver mail. In 1506, Michele was murdered by robbers, and Nicolo, his two siblings, and his mother were left impoverished. Nicolo experienced further tragedy in 1512 when King Louis XII's troops invaded Brescia during the War of the League of Cambrai against Venice. The militia of Brescia defended their city for seven days. When the French finally broke through, they took their revenge by massacring the inhabitants of Brescia. By the end of battle, over 45,000 residents were killed. During the massacre, Nicolo and his family sought sanctuary in the local cathedral. But the French entered and a soldier sliced Nicolo's jaw and palate with a saber and left him for dead. His mother nursed him back to health but the young boy was left with a speech impediment, prompting the nickname "Tartaglia" ("stammerer"). After this he would never shave, and grew a beard to camouflage his scars. His surname at birth, if any, is disputed. Some sources have him as "Niccolò Fontana", but others claim that the only support for this is a will in which he named a brother, Zuampiero Fontana, as heir, and point out that this does not imply he had the same surname. Tartaglia's biographer Arnoldo Masotti writes that: At the age of about fourteen, he [Tartaglia] went to a Master Francesco to learn to write the alphabet; but by the time he reached “k,” he was no longer able to pay the teacher. “From that day,” he later wrote in a moving autobiographical sketch, “I never returned to a tutor, but continued to labour by myself over the works of dead men, accompanied only by the daughter of poverty that is called industry” (Quesiti, bk. VI, question 8). Tartaglia moved to Verona around 1517, then to Venice in 1534, a major European commercial hub and one of the great centres of the Italian renaissance at this time. Also relevant is Venice's place at the forefront of European printing culture in the sixteenth century, making early printed texts available even to poor scholars if sufficiently motivated or well-connected — Tartaglia knew of Archimedes' work on the quadrature of the parabola, for example, from Guarico's Latin edition of 1503, which he had found "in the hands of a sausage-seller in Verona in 1531" (in mano di un salzizaro in Verona, l'anno 1531 in his words). Tartaglia's mathematics is also influenced by the works of medieval Islamic scholar Muhammad ibn Musa Al-Khwarizmi from 12th Century Latin translations becoming available in Europe. Tartaglia eked out a living teaching practical mathematics in abacus schools and earned a penny where he could: This remarkable man [Tartaglia] was a self-educated mathematics teacher who sold mathematical advice to gunners and architects, ten pennies one question, and had to litigate with his customers when they gave him a worn-out cloak for his lectures on Euclid instead of the payment agreed on. He died in Venice. == Ballistics == Nova Scientia (1537) was Tartaglia's first published work, described by Matteo Valleriani as: ... one of the most fundamental works on mechanics of the Renaissance, indeed, the first to transform aspects of practical knowledge accumulated by the early modern artillerists into a theoretical and mathematical framework. Then dominant Aristotelian physics preferred categories like "heavy" and "natural" and "violent" to describe motion, generally eschewing mathematical explanations. Tartaglia brought mathematical models to the fore, "eviscerat[ing] Aristotelian terms of projectile movement" in the words of Mary J. Henninger-Voss. One of his findings was that the maximum range of a projectile was achieved by directing the cannon at a 45° angle to the horizon. Tartaglia's model for a cannonball's flight was that it proceeded from the cannon in a straight line, then after a while started to arc towards the earth along a circular path, then finally dropped in another straight line directly towards the earth. At the end of Book 2 of Nova Scientia, Tartaglia proposes to find the length of that initial rectilinear path for a projectile fired at an elevation of 45°, engaging in a Euclidean-style argument, but one with numbers attached to line segments and areas, and eventually proceeds algebraically to find the desired quantity (procederemo per algebra in his words). Mary J. Henninger-Voss notes that "Tartaglia's work on military science had an enormous circulation throughout Europe", being a reference for common gunners into the eighteenth century, sometimes through unattributed translations. He influenced Galileo as well, who owned "richly annotated" copies of his works on ballistics as he set about solving the projectile problem once and for all. == Translations == Archimedes' works began to be studied outside the universities in Tartaglia's day as exemplary of the notion that mathematics is the key to understanding physics, Federigo Commandino reflecting this notion when saying in 1558 that "with respect to geometry no one of sound mind could deny that Archimedes was some god". Tartaglia published a 71-page Latin edition of Archimedes in 1543, Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi, containing Archimedes' works on the parabola, the circle, centres of gravity, and floating bodies. Guarico had published Latin editions of the first two in 1503, but the works on centres of gravity and floating bodies had not been published before. Tartaglia published Italian versions of some Archimedean texts later in life, his executor continuing to publish his translations after his death. Galileo probably learned of Archimedes' work through these widely disseminated editions. Tartaglia's Italian edition of Euclid in 1543, Euclide Megarense philosopho, was especially significant as the first translation of the Elements into any modern European language. For two centuries Euclid had been taught from two Latin translations taken from an Arabic source; these contained errors in Book V, the Eudoxian theory of proportion, which rendered it unusable. Tartaglia's edition was based on Zamberti's Latin translation of an uncorrupted Greek text, and rendered Book V correctly. He also wrote the first modern and useful commentary on the theory. This work went through many editions in the sixteenth century and helped diffuse knowledge of mathematics to a non-academic but increasingly well-informed literate and numerate public in Italy. The theory became an essential tool for Galileo, as it had been for Archimedes. == General Trattato di Numeri et Misure == Tartaglia exemplified and eventually transcended the abaco tradition that had flourished in Italy since the twelfth century, a tradition of concrete commercial mathematics taught at abacus schools maintained by communities of merchants. Maestros d'abaco like Tartaglia taught not with the abacus but with paper-and-pen, inculcating algorithms of the type found in grade schools today. Tartaglia's masterpiece was the General Trattato di Numeri et Misure (General Treatise on Number and Measure), a 1500-page encyclopedia in six parts written in the Venetian dialect, the first three coming out in 1556 about the time of Tartaglia's death and the last three published posthumously by his literary executor and publisher Curtio Troiano in 1560. David Eugene Smith wrote of the General Trattato that it was: the best treatise on arithmetic that appeared in Italy in his century, containing a very full discussion of the numerical operations and the commercial rules of the Italian arithmeticians. The life of the people, the customs of the merchants, and the efforts at improving arithmetic in the 16th century are all set forth in this remarkable work. Part I is 554 pages long and constitutes essentially commercial arithmetic, taking up such topics as basic operations with the complex currencies of the day (ducats, soldi, pizolli, and so on), exchanging currencies, calculating interest, and dividing profits into joint companies. The book is replete with worked examples with much emphasis on methods and rules (that is, algorithms), all ready to use virtually as is. Part II takes up more general arithmetic problems, including progressions, powers, binomial expansions, Tartaglia's triangle (also known as "Pascal's triangle"), calculations with roots, and proportions / fractions. Part IV concerns triangles, regular polygons, the Platonic solids, and Archimedean topics like the quadrature of the circle and circumscribing a cylinder around a sphere. == Tartaglia's triangle == Tartaglia was proficient with binomial expansions and included many worked examples in Part II of the General Trattato, one a detailed explanation of how to calculate the summands of ( 6 + 4 ) 7 {\displaystyle (6+4)^{7}} , including the appropriate binomial coefficients. Tartaglia knew of Pascal's triangle one hundred years before Pascal, as shown in this image from the General Trattato. His examples are numeric, but he thinks about it geometrically, the horizontal line a b {\displaystyle ab} at the top of the triangle being broken into two segments a c {\displaystyle ac} and c b {\displaystyle cb} , where point c {\displaystyle c} is the apex of the triangle. Binomial expansions amount to taking ( a c + c b ) n {\displaystyle (ac+cb)^{n}} for exponents n = 2 , 3 , 4 , ⋯ {\displaystyle n=2,3,4,\cdots } as you go down the triangle. The symbols along the outside represent powers at this early stage of algebraic notation: c e = 2 , c u = 3 , c e . c e = 4 {\displaystyle ce=2,cu=3,ce.ce=4} , and so on. He writes explicitly about the additive formation rule, that (for example) the adjacent 15 and 20 in the fifth row add up to 35, which appears beneath them in the sixth row. == Solution to cubic equations == Tartaglia is perhaps best known today for his conflicts with Gerolamo Cardano. In 1539, Cardano cajoled Tartaglia into revealing his solution to the cubic equations by promising not to publish them. Tartaglia divulged the secrets of the solutions of three different forms of the cubic equation in verse. Several years later, Cardano happened to see unpublished work by Scipione del Ferro who independently came up with the same solution as Tartaglia. (Tartaglia had previously been challenged by del Ferro's student Fiore, which made Tartaglia aware that a solution existed.) As the unpublished work was dated before Tartaglia's, Cardano decided his promise could be broken and included Tartaglia's solution in his next publication. Even though Cardano credited his discovery, Tartaglia was extremely upset and a famous public challenge match resulted between himself and Cardano's student, Ludovico Ferrari. Widespread stories that Tartaglia devoted the rest of his life to ruining Cardano, however, appear to be completely fabricated. Mathematical historians now credit both Cardano and Tartaglia with the formula to solve cubic equations, referring to it as the "Cardano–Tartaglia formula". == Volume of a tetrahedron == Tartaglia was a prodigious calculator and master of solid geometry. In Part IV of the General Trattato he shows by example how to calculate the height of a pyramid on a triangular base, that is, an irregular tetrahedron. The base of the pyramid is a 13-14-15 triangle bcd, and the edges rising to the apex a from points b, c, and d have respective lengths 20, 18, and 16. The base triangle bcd partitions into 5-12-13 and 9-12-15 triangles by dropping the perpendicular from point d to side bc. He proceeds to erect a triangle in the plane perpendicular to line bc through the pyramid's apex, point a, calculating all three sides of this triangle and noting that its height is the height of the pyramid. At the last step, he applies what amounts to this formula for the height h of a triangle in terms of its sides p, q, r (the height from side p to its opposite vertex): h 2 = r 2 − ( p 2 + r 2 − q 2 2 p ) 2 , {\displaystyle h^{2}=r^{2}-\left({\frac {p^{2}+r^{2}-q^{2}}{2p}}\right)^{2},} a formula deriving from the law of cosines (not that he cites any justification in this section of the General Trattato). Tartaglia drops a digit early in the calculation, taking 305+31/49 as 305+3/49, but his method is sound. The final (correct) answer is: height of pyramid = 240 615 3136 . {\displaystyle {\text{height of pyramid}}={\sqrt {240{\tfrac {615}{3136}}}}.} The volume of the pyramid is easily obtained from this, though Tartaglia does not give it: V = 1 3 × base × height = 1 3 × Area ( △ b c d ) × height = 1 3 × 84 × 240 615 3136 ≈ 433.9513222 {\displaystyle {\begin{aligned}V&={\tfrac {1}{3}}\times {\text{base}}\times {\text{height}}\\&={\tfrac {1}{3}}\times {\text{Area}}(\triangle bcd)\times {\text{height}}\\&={\tfrac {1}{3}}\times 84\times {\sqrt {240{\tfrac {615}{3136}}}}\\&\approx 433.9513222\end{aligned}}} Simon Stevin invented decimal fractions later in the sixteenth century, so the approximation would have been foreign to Tartaglia, who always used fractions. His approach is in some ways a modern one, suggesting by example an algorithm for calculating the height of irregular tetrahedra, but (as usual) he gives no explicit general formula. == Works == Tartaglia, Niccolò, General Trattato di Numeri et Misure, Part I (Venice, 1556) Tartaglia, Niccolò, General Trattato di Numeri et Misure, Part II (Venice, 1556) Tartaglia, Niccolò, General Trattato di Numeri et Misure, Part III (Venice, 1556) Tartaglia, Niccolò, General Trattato di Numeri et Misure, Part IV (Venice, 1560) Tartaglia, Niccolò, General Trattato di Numeri et Misure, Part V (Venice, 1560) Tartaglia, Niccolò, General Trattato di Numeri et Misure, Part VI (Venice, 1560) == Notes == == References == Chisholm, Hugh, ed. (1911). "Tartaglia, Niccolò" . Encyclopædia Britannica. Vol. 26 (11th ed.). Cambridge University Press. Clagett, Marshall (1982). "William of Moerbeke: Translator of Archimedes". Proceedings of the American Philosophical Society. 126 (5): 356–366.. Henninger-Voss, Mary J. (July 2002). "How the 'New Science' of Cannons Shook up the Aristotelian Cosmos". Journal of the History of Ideas. 63 (3): 371–397. doi:10.1353/jhi.2002.0029. S2CID 170464547. Herbermann, Charles, ed. (1913). "Nicolò Tartaglia" . Catholic Encyclopedia. New York: Robert Appleton Company. Charles Hutton (1815). "Tartaglia or Tartaglia (Nicholas)". A philosophical and mathematical dictionary. Printed for the author. p. 482. Katz, Victor J. (1998), A History of Mathematics: An Introduction (2nd ed.), Reading: Addison Wesley Longman, ISBN 0-321-01618-1. Malet, Antoni (2012). "Euclid's Swan Song: Euclid's Elements in Early Modern Europe". In Olmos, Paula (ed.). Greek Science in the Long Run: Essays on the Greek Scientific Tradition (4th c. BCE-17th c. CE). Cambridge Scholars Publishing. pp. 205–234. ISBN 978-1-4438-3775-0.. Masotti, Arnoldo (1970). "Niccolò Tartaglia". In Gillispie, Charles (ed.). Dictionary of Scientific Biography. New York: Scribner & American Council of Learned Societies. Smith, D.E. (1958), History of Mathematics, vol. I, New York: Dover Publications, ISBN 0-486-20429-4 {{citation}}: ISBN / Date incompatibility (help). Strathern, Paul (2013), Venetians, New York, NY: Pegasus Books. Tartaglia, Niccolò (1543). Opera Archimedis Syracusani philosophi et mathematici ingeniosissimi. Venice. Tartaglia, Niccolò (1543). Euclide Megarense philosopho. Venice. Tartaglia, Niccolò (1556–1560), General Trattato di Numeri et Misure, Venice: Curtio Troiano. Valleriani, Matteo (2013), Metallurgy, Ballistics and Epistemic Instruments: The Nova Scientia of Nicolò Tartaglia, Berlin: Edition Open Access / Max Planck Research Library, ISBN 978-3-8442-5258-3. Zilsel, Edgar (2000), Raven, Diederick; Krohn, Wolfgang; Cohen, Robert S. (eds.), The Social Origins of Modern Science, Springer Netherlands, ISBN 0-7923-6457-0. == Further reading == Valleriani, Matteo, Metallurgy, Ballistics and Epistemic Instruments: The Nova scientia of Nicolò Tartaglia == External links == History Today Archived 22 January 2012 at the Wayback Machine The Galileo Project O'Connor, John J.; Robertson, Edmund F., "Nicolo Tartaglia", MacTutor History of Mathematics Archive, University of St Andrews Tartaglia's work (and poetry) on the solution of the Cubic Equation at Convergence La Nova Scientia (Venice, 1550)
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Wikipedia:Nicos Christofides#0
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Nicos Christofides (born 1942 in Cyprus; died 2019) was a Cypriot mathematician and professor of financial mathematics at Imperial College London. Christofides studied electrical engineering at Imperial College London, where he also received his PhD in 1966 (dissertation: The origin of load losses in induction motors with cast aluminum rotors). He was briefly with Associated Electrical Industries and then again at Imperial College. In 1976, he devised the Christofides algorithm, an algorithm for finding approximate solutions to the travelling salesman problem. The Christofides algorithm is considered "groundbreaking" and has collected over 2200 citations. In 1982, he rejoined Imperial College London as a professor of operations research. In 1990, he was the co-founder and director of the Centre for Quantitative Finance (now the Institute for Financial Engineering). Christofides became Professor Emeritus of Quantitative Finance at Imperial College London in 2009. He died in 2019. == References ==
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Wikipedia:Nicușor Dan#0
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Nicușor Daniel Dan (Romanian: [nikuˈʃor daniˈel dan]; born 20 December 1969) is a Romanian politician, mathematician, and civic activist who is the president-elect of Romania. He has served as the Mayor of Bucharest since 2020. Born in Făgăraș, Brașov County, Dan earned international acclaim in his youth as a mathematician, securing gold medals at the 1987 and 1988 International Mathematical Olympiads. He began studying mathematics at the University of Bucharest and then moved to France, where he obtained a master's degree from École normale supérieure and a PhD from Paris 13 University. After returning to Romania, Dan founded Școala Normală Superioară București, an institution aimed at guiding the most talented Romanian students towards scientific research, and became a civic activist. In 2015, Dan launched the Save Bucharest Union, championing anti-corruption and heritage preservation, which propelled him into politics. One year later, he co-founded the Save Romania Union (USR), but resigned from the party in 2017 over its progressive shift, preferring a more centrist approach. Dan served in the Chamber of Deputies from 2016, before being elected Bucharest’s first independent mayor in 2020 and winning re-election in 2024. He is focused on public infrastructure and transparency, despite criticism over construction delays. Dan ran as an independent for the 2025 Romanian presidential election and received 21% of the vote in the first round. He faced Alliance for the Union of Romanians (AUR) founder George Simion in the runoff, defeating him with 53.6% of the vote. Dan's staunch pro-Western platform contrasted with his opponent's nationalist and Eurosceptic stance. == Early life and education == Born in Făgăraș, Brașov County, he attended the Radu Negru High School in his native city, graduating in 1988. He won first prizes in the International Mathematical Olympiads in 1987 and 1988 with perfect scores. Dan moved to Bucharest at the age of 18 and began studying mathematics at the University of Bucharest. In 1992, he moved to France to continue studying mathematics: he followed the courses of the École Normale Supérieure, one of the most prestigious French grandes écoles, where he gained a master's degree. In 1998 Dan completed a PhD in mathematics at Paris 13 University, with thesis "Courants de Green et prolongement méromorphe" written under the direction of Christophe Soulé and Daniel Barsky. He returned to Bucharest that year, giving as reasons the cultural differences and the desire to change Romania. Dan was one of the creators and the first administrative director of the Școala Normală Superioară București, a university set up on the model of the French École Normale Supérieure within the Romanian Academy's Institute of Mathematics. As of 2011, he was a professor of mathematics at the institute. == Activism == In 1998, Dan founded Asociația "Tinerii pentru Acțiune Civică" ("Young People for Civic Action" Association), for which he wanted to gather a thousand young people who wanted to change Romania, which was his stated goal for returning to the country. Despite failing in its goals, the association did organise two forums for young people who studied abroad, in 2000 and 2002, to which a few hundred people participated. As result of these forums, the "Ad Astra" Association of Romanian researchers was created in 2000. === Save Bucharest Association === Dan founded the Asociația "Salvați Bucureștiul" ("Save Bucharest" Association) in 2006 as a reaction to the demolition of architectural heritage houses and the building of high-rise buildings in protected Bucharest neighborhoods, as well as the diminishing number of green space areas in Bucharest. In March 2008, the association published the "Bucharest, an urbanistic disaster" Report, which discussed Bucharest's problems and ways to overcome them. In the same year, during the elections, together with other NGOs, the association drafted a Pact for Bucharest, which was signed by all the candidates for mayor of Bucharest. On April Fools' Day in 2012, Dan published a list of 100 electoral promises made by elected mayor of Bucharest Sorin Oprescu which were not kept, including the "Pact for Bucharest". The association was involved in many trials, winning 23 trials against the local authorities of Bucharest. Among them are the cancellation of a project which would have built a water park on 7 hectares of Tineretului Park, saving from demolition a number of heritage buildings on Șoseaua Kiseleff no. 45, and the cancelation of a project which would have built a glass building on top of Palatul Știrbei on Calea Victoriei. The association was also able to push some changes in 2009 to the urban planning law. == Early political career == === 2012 local elections === Dan announced his candidacy for Mayor of Bucharest in November 2011 at a café on Arthur Verona Street, with just a few guests, among which Theodor Paleologu, a historian and Member of Parliament. For gathering the 36,000 signatures needed for his candidacy, having the backing of no party, he relied on a network of volunteers organised on Facebook. On 22 April, 15 bands and musicians performed pro-bono at Arenele Romane for Dan's campaign in order to help him gather the signatures. During the 12-hour-long concert, volunteers gathered 4,000 signatures. ==== Political positions and programme ==== Among his proposed projects are the creation of a light rail infrastructure over the existing rail lines in Bucharest, creating an infrastructure for prioritising public transport over other traffic in intersections, consolidating buildings that are likely to be affected by earthquakes, protecting the urban green space and clearing illegal buildings from parks. Dan argues that it is important to incentivise young people to stay in the city, by making it a regional hub in IT, creative industries and higher education, and attracting investors and skilled people from across the region. ==== Support and opinions on his candidacy ==== He received support from Andrei Pleșu, who argued that Dan is the only one of the candidates who is interested in the architecture of Bucharest and does not support any utopian initiatives. He also received support from political scientist and Member of the European Parliament Cristian Preda. Dan gained the support of some journalists who wrote about him in op-eds from several newspapers: Andrei Crăciun of Adevărul saw in him "a Don Quijote untouched by the vulgar lard of undeserved riches" and "a person who works against the system". Florin Negruțiu, the editor-in-chief of Gândul thought he is an "atypical candidate" for Bucharest, the model candidate of the intellectuals; nevertheless, the journalist did not see any chances that Dan would become mayor, because he is "too serious" a candidate, and unlikely to appeal to the masses. Neculai Constantin Munteanu from Radio Free Europe wrote that he supports Dan for his unselfish way of caring about Bucharest and that his opponents are "comedians", for which one can "admire the imposture, ludicrousness, and incompetence". === 2016 local elections === Having registered Save Bucharest Union (USB) as a political party in 2015, Dan ran again for Mayor of Bucharest in 2016. This time, the elections were held in a single round. He gained 30,52% of the total votes, losing to the socialist candidate, Gabriela Firea, who gained 42,97% of the total votes. In the election, Dan managed to attract the young electorate, with over half of his voters being under the age of 40. Some of USB's candidates for sector mayor have also performed well in their respective races, proving USB's viability as a future political force. === Save Romania Union === Wanting to capitalise on the momentum that saw him gain a third of the votes in the local elections, Dan announced shortly after the 2016 local election that the Save Bucharest Union will change its name to Save Romania Union (USR), shifting its focus to a national stage. He also announced plans for the new party to enter the parliamentary elections of that year. With Dan at the top of the candidate list, USR gained 8.92% of the vote in the Senate race and 8.87% in the Chamber of Deputies, which made them the third largest party in Romania. The result also meant that Dan became a member of the Chamber of Deputies. ==== Departure from USR ==== In 2017, anti-same-sex NGO Coaliția pentru Familie managed to raise the necessary number of signatures to organise a referendum that would change the part of the Romanian Constitution dealing with marriage, with the hope of redefining it as "between a man and a woman". This created a rift within USR, between the progressive wing, who wanted USR to become the only parliamentary party to oppose the initiative, and Dan, who believed USR should not get involved in the debate and that the party should remain open for both progressives and conservatives. An internal referendum within the party followed, in which 52.7% of members voted to position the party against the Constitutional initiative, which led Dan to resign from the party on 1 June 2017. As explanation for his opposition to the National Council vote he cited religious matters, the dangers of deviating from the main party issue of fighting against corruption and his refusal to belong to a party that defines itself as a party of civil liberties. === Independent === After his resignation from USR, Dan continued to serve as a member of the Chamber of Deputies as an independent. Due to a quirk in the Romanian electoral law, USR required his signature when they attempted to legally register their alliance with the Freedom, Unity and Solidarity Party (PLUS). In order to help his former party, in March 2019 Dan briefly rejoined USR as a common member, gave the necessary signature and then resigned for a second time. == Mayor of Bucharest (2020–2025) == === 2020 local elections === In May 2019, he announced his plans to once again run for Mayor of Bucharest, as an independent. Dan mentioned that while he hoped that his candidacy would be supported by the rest of the opposition parties, he would not run against a different common candidate, unwilling to split the vote of the opposition. He was ultimately supported by both USR and the National Liberal Party (PNL). With 95% of votes counted, partial results suggested that he won the mayoral election with 42.8% of votes. Shortly afterwards exit polls showed him winning the race, he announced victory. On 5 October 2020 the Central Electoral Bureau confirmed his status as the new Mayor of Bucharest, winning the elections with a plurality of 42.81% against Gabriela Firea (37.97%), the former Mayor. === 2024 local elections === Following the decision made by the governing alliance between the National Liberal Party (PNL) and the Social Democratic Party (PSD) to hold the elections in June of 2024, Dan participated once again as an independent for Mayor, for a new term. This time, he was supported by the same USR (Save Romanian Union) party, but also by two other minor parties, the People's Movement Party (PMP) and The Force of the Right (FD) whose president is former PNL leader Ludovic Orban, who left the party in 2021 after losing the presidency of the party to then-prime minister Florin Cîțu; all three formed the United Right Alliance (ADU), an official national opposition to the National Coalition for Romania (CNR) formed by the PSD and PNL. Additionally, the REPER party, headed by former PLUS leader Dacian Cioloș, supported Dan, but was not part of ADU. The elections were held on 9 June 2024 together with the European Parliament elections in Romania, a controversial move done by the CNR earlier that year. Thought to be a close race up until the last moment, the exit polls showed the result was overwhelmingly in favour of Dan, winning with 45% of the total vote, who declared himself the winner of the race. After the vote count, Dan was the clear winner of the elections with approximately 48% of the total votes, more than double the votes given to the same runner-up from 2020, Gabriela Firea, who placed second with 22%, followed by former Sector 5 Mayor, Cristian Popescu Piedone (16%) and PNL candidate and president for the Bucharest branch of the party, Sebastian Burduja (7.6%). During his victory speech, Dan declared his intention to organise two referendums for Bucharest, one for centralising more power to the General Mayor of Bucharest regarding building authorisations, a very consistent theme during his campaign, and another for allocating more financial funds to the General Mayor rather than to the Sector mayors. Both were planned to take place on the same day as the parliamentary elections, in order to "reduce organisational costs for separate elections", according to Dan. ==== Piața Unirii incident ==== On 14 October 2024, around midnight, Sector 4 mayor Daniel Băluță (PSD), declared that the foundation upon which the central Piața Unirii stands has become a public danger due to its age and sent multiple construction workers and Sector 4 local police agents to Unirii Park in order to start proceedings for the foundation's physical consolidation. Having learned of this, in the morning of the same day Dan went to the square in person, together with his staff, and Bucharest local police agents, telling the workers present to halt the procedure on the basis of its alleged illegality due to a lack of permits. It was claimed the Sector Mayor did not wait to get all construction permits to proceed with work on the foundation, including permissions from Metrorex and Apa Nova (Bucharest's water and sewage administration institution involved due the fact that the Dâmbovița's courses through the centre of the Piața Unirii) and it was alleged that the sector mayor intended to rush the work in order to embezzle funds allocated to them. Once Dan and his staff made their appearance at the site of the construction works, Sector 4 local police agents including its director, Cristian Pîslă, who was subsequently suspected of corruption, did not permit the entrance to the site itself, and a small scuffle ensued in which the crowds pushed each other. Afterwards, Pîslă called 112, accusing the Mayor of inciting to violence and illegal behaviour. Dan himself called the National Police and after a few more exchanges, left the park in order to retrieve certain documents attesting to the fact that the City Hall of Bucharest had the absolute right to investigate the construction as it was its property, and not Sector 4's. Returning with the documents and some bulldozers, Dan was set to bring down the fences around the site and dismantle the construction works. Sector 4 local police again blocked the entrance of the bulldozers, some agents hurling insults and being physically aggressive to the bulldozer operators, which the wide public of Bucharest viewed as proof of the agents being members of the Sector 4 Clanul Sportivilor, an organisation of the Romanian mafia operating mainly in the southern part of Bucharest who were long suspected to work with Daniel Băluță himself. Reportedly, two people were hurt in the chaos, but these reports were widely ignored as they were viewed as fake in order to pin blame on Dan. Eventually, Daniel Băluță conceded and told the construction workers and police agents to retreat, as Romanian Prime-minister Marcel Ciolacu himself intervened. The construction site was dismantled the following day and Dan launched an investigation into the proceedings. He once again began talking about the referendum for restricting the authority of the sector mayors and centralising more power to the Mayor of Bucharest, a promise made after his re-election in the local elections, and said the General Council of the City Hall of Bucharest will convene on 21 October in order to announce the subsequent date of the referendum. == Presidency (2025–present) == After the Piața Unirii incident, Dan was viewed even more favourably by the general populace of Bucharest, being called a bulwark against the widespread corruption of the country and the only one to effectively stand against the PSD-PNL coalition. This led to speculation of a possible presidential candidature in the next elections. On 16 December he announced his candidacy for the 2025 Romanian presidential election, after the annulment of the 2024 elections due to Russian meddling in favour of winner of the first round Călin Georgescu. His announcement came as a surprise to many, as he had previously expressed his intention to serve at least one more term as Mayor of Bucharest before the elections, stating that he "would need at least 2-3 terms to make everything right in Bucharest. This change of plans also led to a falling out with Elena Lasconi, a former supporter of Dan, who came second in the annulled 2024 elections' first round. It is widely believed that Lasconi and Dan appeal to similar voter demographics, with both targeting liberal, progressive, moderate, pro-European, and anti-PSD/anti-PNL camps. As a result, their simultaneous candidacies may have divided this voter base. The Constitutional Court validated his candidacy on 16 March along with the ones of George Simion and Victor Ponta. On March 22, a random draw placed Dan at the bottom of the candidate list on the ballot. Dan came second in the first round of voting on 4 May with 20.99% of the vote. On 18 May, he faced George Simion in a runoff, winning the presidency with 53.6% of the vote. == Electoral history == === Mayor of Bucharest === === Presidential elections === == Political positions == === Geopolitical alignment === Dan has consistently advocated pro-Western views, emphasizing Romania's integration into the European Union and NATO as cornerstones of national security and economic progress. In his 2025 presidential campaign, Dan positioned himself as a staunch defender of Western democratic values; he supported NATO's presence in Romania, particularly amid the Russian invasion of Ukraine, contrasting sharply with George Simion's nationalism and Euroscepticism. However, Dan faced accusations over his ties with businessman Matei Păun, who was linked to Russian and Belarusian firms and oligarchs. Păun's firm, BAC Financial Advisory SRL, acquired Getica OOH in 2011 from News Corp via Russian VTB Bank and Alpha Capital Partners. Păun allegedly boasted of financing Belarusian leader Alexander Lukashenko and made statements questioning 2014 Western sanctions on Russia, praising Russian Orthodox "mysticism", and doubting Ukraine's Orange Revolution and Crimea's annexation. === LGBT rights === In 2000, Dan published an article in the magazine Dilema in which he stated his rejection of "homosexual behaviour in public spaces in Romania," describing it as "an attack against traditional values" and "legitimate collective identity". The statements resurfaced after his political career took off, particularly during his much-publicized departure from USR. Dan distanced himself from his previous statements on several occasions, claiming that he is not homophobic and that his opinion on the matter has changed in the following years. === Alleged Securitate collaboration === In May 2024, ahead of local elections, a purported Securitate document from July 1988 emerged, detailing Dan's collaboration with the secret police of the Ceaușescu regime. The document contained information provided by Dan about his high school peers who participated in the International Mathematical Olympiads of 1987 and 1988. Dan denied its authenticity, claiming he had minimal contact with authorities of the time. PSD leader and Prime Minister Marcel Ciolacu questioned the document's credibility, noting its unusually polished composition. The National Council for the Study of the Securitate Archives (CNSAS) deemed it a forgery, citing incorrect dates, atypical expressions, and a writing style inconsistent with the typewriters used by the Securitate, including the absence of diacritics typical in genuine documents of the era. == Personal life == Dan resides with his long-term partner, Mirabela, a Renault executive. They welcomed a daughter in May 2016 and a son in May 2022. == References == == External links == Nicușor Dan's website Salvați Bucureștiul (Save Bucharest) Association Păun, Carmen (3 June 2016). "Geek takes on Romanian establishment. The unlikely campaign of a mathematician dedicated to saving Bucharest from the developers — and the politicians". Politico. Retrieved 27 September 2020. VIAF 45153124238924490799
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Wikipedia:Niels Fabian Helge von Koch#0
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Niels Fabian Helge von Koch (25 January 1870 – 11 March 1924) was a Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake, one of the earliest fractal curves to be described. He was born to Swedish nobility. His grandfather, Nils Samuel von Koch (1801–1881), was the Chancellor of Justice. His father, Richert Vogt von Koch (1838–1913) was a Lieutenant-Colonel in the Life Guards of Horse of the Swedish Army. He was enrolled at the newly created Stockholm University College in 1887 (studying under Gösta Mittag-Leffler), and at Uppsala University in 1888, where he also received his bachelor's degree (filosofie kandidat) since the non-governmental college in Stockholm had not yet received the rights to issue degrees. He received his PhD in Uppsala in 1892. He was appointed professor of mathematics at the Royal Institute of Technology in Stockholm in 1905, succeeding Ivar Bendixson, and became professor of pure mathematics at Stockholm University College in 1911. Von Koch wrote several papers on number theory. One of his results was a 1901 theorem proving that the Riemann hypothesis implies what is now known to be the strongest possible form of the prime number theorem. He described the Koch curve in a 1904 paper entitled Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire ("On a continuous curve without tangents constructible from elementary geometry"). He was an invited speaker at the International Congress of Mathematicians in 1900 in Paris with talk Sur la distribution des nombres premiers ("On the distribution of prime numbers") and in 1912 in Cambridge, England, with talk On regular and irregular solutions of some infinite systems of linear equations. == Notes == == References == Marquis of Ruvigny and Raineval (1911). The Plantagenet Roll of the Blood Royal. Vol. Mortimer–Percy. pp. 250–251. Edgar, Gerald, ed. (1993). Classics on Fractals. Addison-Wesley. ISBN 9780201587012. contains an English translation of the paper 'On a continuous curve...' == External links == Works by or about Niels Fabian Helge von Koch at the Internet Archive O'Connor, John J.; Robertson, Edmund F., "Niels Fabian Helge von Koch", MacTutor History of Mathematics Archive, University of St Andrews Niels Fabian Helge von Koch at the Mathematics Genealogy Project von Koch family
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Wikipedia:Niels Henrik Abel#0
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Niels Henrik Abel ( AH-bəl, Norwegian: [ˌnɪls ˈhɛ̀nːɾɪk ˈɑ̀ːbl̩]; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solving the general quintic equation in radicals. This question was one of the outstanding open problems of his day, and had been unresolved for over 250 years. He was also an innovator in the field of elliptic functions and the discoverer of Abelian functions. He made his discoveries while living in poverty and died at the age of 26 from tuberculosis. Most of his work was done in six or seven years of his working life. Regarding Abel, the French mathematician Charles Hermite said: "Abel has left mathematicians enough to keep them busy for five hundred years." Another French mathematician, Adrien-Marie Legendre, said: "What a head the young Norwegian has!" == Life == === Early life === Niels Henrik Abel was born prematurely in Nedstrand, Norway, as the second child of the pastor Søren Georg Abel and Anne Marie Simonsen. When Niels Henrik Abel was born, the family was living at a rectory on Finnøy. Much suggests that Niels Henrik was born in the neighboring parish, as his parents were guests of the bailiff in Nedstrand in July / August of his year of birth. Niels Henrik Abel's father, Søren Georg Abel, had a degree in theology and philosophy and served as pastor at Finnøy. Søren's father, Niels's grandfather, Hans Mathias Abel, was also a pastor, at Gjerstad Church near the town of Risør. Søren had spent his childhood at Gjerstad, and had also served as chaplain there; and after his father's death in 1804, Søren was appointed pastor at Gjerstad and the family moved there. The Abel family originated in Schleswig and came to Norway in the 17th century. Anne Marie Simonsen was from Risør; her father, Niels Henrik Saxild Simonsen, was a tradesman and merchant ship-owner, and said to be the richest person in Risør. Anne Marie had grown up with two stepmothers, in relatively luxurious surroundings. At Gjerstad rectory, she enjoyed arranging balls and social gatherings. Much suggests she was early on an alcoholic and took little interest in the upbringing of the children. Niels Henrik and his brothers were given their schooling by their father, with handwritten books to read. An addition table in a book of mathematics reads: 1+0=0. === Cathedral School and Royal Frederick University === With Norwegian independence and the first election held in Norway, in 1814, Søren Abel was elected as a representative to the Storting. Meetings of the Storting were held until 1866 in the main hall of the Cathedral School in Christiania (now known as Oslo). Almost certainly, this is how he came into contact with the school, and he decided that his eldest son, Hans Mathias, should start there the following year. However, when the time for his departure approached, Hans was so saddened and depressed over having to leave home that his father did not dare send him away. He decided to send Niels instead. In 1815, Niels Abel entered the Cathedral School at the age of 13. His elder brother Hans joined him there a year later. They shared rooms and had classes together. Hans got better grades than Niels; however, a new mathematics teacher, Bernt Michael Holmboe, was appointed in 1818. He gave the students mathematical tasks to do at home. He saw Niels Henrik's talent in mathematics, and encouraged him to study the subject to an advanced level. He even gave Niels private lessons after school. In 1818, Søren Abel had a public theological argument with the theologian Stener Johannes Stenersen regarding his catechism from 1806. The argument was well covered in the press. Søren was given the nickname "Abel Treating" (Norwegian: "Abel Spandabel"). Niels' reaction to the quarrel was said to have been "excessive gaiety". At the same time, Søren also almost faced impeachment after insulting Carsten Anker, the host of the Norwegian Constituent Assembly; and in September 1818 he returned to Gjerstad with his political career in ruins. He began drinking heavily and died only two years later, in 1820, aged 48. Bernt Michael Holmboe supported Niels Henrik Abel with a scholarship to remain at the school and raised money from his friends to enable him to study at the Royal Frederick University. When Abel entered the university in 1821, he was already the most knowledgeable mathematician in Norway. Holmboe had nothing more he could teach him and Abel had studied all the latest mathematical literature in the university library. During that time, Abel started working on the quintic equation in radicals. Mathematicians had been looking for a solution to this problem for over 250 years. In 1821, Abel thought he had found the solution. The two professors of mathematics in Christiania, Søren Rasmussen and Christopher Hansteen, found no errors in Abel's formulas, and sent the work on to the leading mathematician in the Nordic countries, Carl Ferdinand Degen in Copenhagen. He too found no faults but still doubted that the solution, which so many outstanding mathematicians had sought for so long, could really have been found by an unknown student in far-off Christiania. Degen noted, however, Abel's unusually sharp mind, and believed that such a talented young man should not waste his abilities on such a "sterile object" as the fifth degree equation, but rather on elliptic functions and transcendence; for then, wrote Degen, he would "discover Magellanian thoroughfares to large portions of a vast analytical ocean". Degen asked Abel to give a numerical example of his method. While trying to provide an example, Abel found a mistake in his paper. This led to a discovery in 1823 that a solution by formula to a fifth- or higher-degree equation was not necessarily possible. Abel graduated in 1822. His performance was exceptionally high in mathematics and average in other matters. === Career === After he graduated, professors from university supported Abel financially, and Professor Christopher Hansteen let him live in a room in the attic of his home. Abel would later view Ms. Hansteen as his second mother. While living here, Abel helped his younger brother, Peder Abel, through examen artium. He also helped his sister Elisabeth to find work in the town. In early 1823, Niels Abel published his first article in "Magazin for Naturvidenskaberne", Norway's first scientific journal, which had been co-founded by Professor Hansteen. Abel published several articles, but the journal soon realized that this was not material for the common reader. In 1823, Abel also wrote a paper in French. It was "a general representation of the possibility to integrate all differential formulas" (Norwegian: en alminnelig Fremstilling af Muligheten at integrere alle mulige Differential-Formler). He applied for funds at the university to publish it. However, the work was lost while being reviewed, never to be found thereafter. In mid-1823, Professor Rasmussen gave Abel a gift of 100 speciedaler so he could travel to Copenhagen and visit Ferdinand Degen and other mathematicians there. While in Copenhagen, Abel did some work on Fermat's Last Theorem. Abel's uncle, Peder Mandrup Tuxen, lived at the naval base in Christianshavn, Copenhagen, and at a ball there Niels Abel met Christine Kemp, his future fiancée. In 1824, Christine moved to Son, Norway, to work as a governess and the couple got engaged over Christmas. After returning from Copenhagen, Abel applied for a government scholarship in order to visit top mathematicians in Germany and France, but he was instead granted 200 speciedaler yearly for two years, to stay in Christiania and study German and French. In the next two years, he was promised a scholarship of 600 speciedaler yearly and he would then be permitted to travel abroad. While studying these languages, Abel published his first notable work in 1824, Mémoire sur les équations algébriques où on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré (Memoir on algebraic equations, in which the impossibility of solving the general equation of the fifth degree is proven). By 1823, Abel had at last proved the impossibility of solving the quintic equation in radicals (now referred to as the Abel–Ruffini theorem). However, this paper was in an abstruse and difficult form, in part because he had restricted himself to only six pages in order to save money on printing. A more detailed proof was published in 1826 in the first volume of Crelle's Journal. In 1825, Abel wrote a personal letter to King Carl Johan of Norway/Sweden requesting permission to travel abroad. He was granted this permission, and in September 1825 he left Christiania together with four friends from university (Christian P.B Boeck, Balthazar M. Keilhau, Nicolay B. Møller and Otto Tank). These four friends of Abel were traveling to Berlin and to the Alps to study geology. Abel wanted to follow them to Copenhagen and from there make his way to Göttingen. The terms for his scholarship stipulated that he was to visit Gauss in Göttingen and then continue to Paris. However, when he got as far as Copenhagen, he changed his plans. He wanted to follow his friends to Berlin instead, intending to visit Göttingen and Paris afterwards. On the way, he visited the astronomer Heinrich Christian Schumacher in Altona, now a district of Hamburg. He then spent four months in Berlin, where he became well acquainted with August Leopold Crelle, who was then about to publish his mathematical journal, Journal für die reine und angewandte Mathematik. This project was warmly encouraged by Abel, who contributed much to the success of the venture. Abel contributed seven articles to it in its first year. From Berlin Abel also followed his friends to the Alps. He went to Leipzig and Freiberg to visit Georg Amadeus Carl Friedrich Naumann and his brother the mathematician August Naumann. In Freiberg Abel did research in the theory of functions, particularly, elliptic, hyperelliptic, and a new class now known as abelian functions. From Freiberg they went on to Dresden, Prague, Vienna, Trieste, Venice, Verona, Bolzano, Innsbruck, Luzern and Basel. From July 1826 Abel traveled on his own from Basel to Paris. Abel had sent most of his work to Berlin to be published in Crelle's Journal, but he had saved what he regarded as his most important work for the French Academy of Sciences, a theorem on addition of algebraic differentials. With the help of a painter, Johan Gørbitz, he found an apartment in Paris and continued his work on the theorem. He finished in October 1826 and submitted it to the academy. It was to be reviewed by Augustin-Louis Cauchy. Abel's work was scarcely known in Paris, and his modesty restrained him from proclaiming his research. The theorem was put aside and forgotten until his death. Abel's limited finances finally compelled him to abandon his tour in January 1827. He returned to Berlin, and was offered a position as editor of Crelle's Journal, but opted out. By May 1827 he was back in Norway. His tour abroad was viewed as a failure. He had not visited Gauss in Göttingen and he had not published anything in Paris. His scholarship was therefore not renewed and he had to take up a private loan in Norges Bank of 200 spesidaler. He never repaid this loan. He also started tutoring. He continued to send most of his work to Crelle's Journal. But in mid-1828 he published, in rivalry with Carl Jacobi, an important work on elliptic functions in Astronomische Nachrichten in Altona. === Death === While in Paris, Abel contracted tuberculosis. At Christmas 1828, he traveled by sled to Froland, Norway, to visit his fiancée. He became seriously ill on the journey. Although a temporary improvement allowed the couple to enjoy the holiday together, he died relatively soon after on 6 April 1829, just two days before a letter arrived from August Crelle telling Abel that Crelle had secured him an appointment as a professor at the University of Berlin. == Contributions to mathematics == Abel showed that there is no general algebraic solution for the roots of a quintic equation, or any general polynomial equation of degree greater than four, in terms of explicit algebraic operations with the Abel-Ruffini theorem. To do this, he invented (independently of Galois) a branch of mathematics known as group theory, which is invaluable not only in many areas of mathematics, but for much of physics as well. Abel sent a paper on the unsolvability of the quintic equation to Carl Friedrich Gauss, who proceeded to discard without a glance what he believed to be the worthless work of a crank. As a 16-year-old, Abel gave a rigorous proof of the binomial theorem valid for all numbers, extending Euler's result which had held only for rationals. Abel wrote a fundamental work on the theory of elliptic integrals, containing the foundations of the theory of elliptic functions. While travelling to Paris he published a paper revealing the double periodicity of elliptic functions, which Adrien-Marie Legendre later described to Augustin-Louis Cauchy as "a monument more lasting than bronze" (borrowing a famous sentence by the Roman poet Horatius). The paper was, however, misplaced by Cauchy. While abroad Abel had sent most of his work to Berlin to be published in the Crelle's Journal, but he had saved what he regarded as his most important work for the French Academy of Sciences, a theorem on addition of algebraic differentials. The theorem was put aside and forgotten until his death. While in Freiberg, Abel did research in the theory of functions, particularly, elliptic, hyperelliptic, and a new class now known as abelian functions. In 1823 Abel wrote a paper titled "a general representation of the possibility to integrate all differential formulas" (Norwegian: en alminnelig Fremstilling af Muligheten at integrere alle mulige Differential-Formler). He applied for funds at the university to publish it. However the work was lost, while being reviewed, never to be found thereafter. Abel said famously of Carl Friedrich Gauss's writing style, "He is like the fox, who effaces his tracks in the sand with his tail." Gauss replied to him by saying, "No self-respecting architect leaves the scaffolding in place after completing his building." == Legacy == Under Abel's guidance, the prevailing obscurities of analysis began to be cleared, new fields were entered upon and the study of functions so advanced as to provide mathematicians with numerous ramifications along which progress could be made. His works, the greater part of which originally appeared in Crelle's Journal, were edited by Bernt Michael Holmboe and published in 1839 by the Norwegian government, and a more complete edition by Ludwig Sylow and Sophus Lie was published in 1881. The adjective "abelian", derived from his name, has become so commonplace in mathematical writing that it is conventionally spelled with a lower-case initial "a" (e.g., abelian group, abelian category, and abelian variety). On 6 April 1929, four Norwegian stamps were issued for the centenary of Abel's death. His portrait appears on the 500-kroner banknote (version V) issued during 1978–1985. On 5 June 2002, four Norwegian stamps were issued in honour of Abel two months before the bicentenary of his birth. There is also a 20-kroner coin issued by Norway in his honour. A statue of Abel stands in Oslo, and crater Abel on the Moon was named after him. The Abel Prize in mathematics was established in Abel's memory and named in his honour. Although it was originally proposed in 1899 to complement the Nobel Prizes, it was first awarded in 2003, while Selberg received an honorary Abel Prize the previous year. Mathematician Felix Klein wrote about Abel: But I would not like to part from this ideal type of researcher, such as has seldom appeared in the history of mathematics, without evoking a figure from another sphere who, in spite of his totally different field, still seems related. Thus, although Abel shared with many mathematicians a complete lack of musical talent, I will not sound absurd if I compare his kind of productivity and his personality with Mozart's. Thus one might erect a monument to this divinely inspired mathematician like the one to Mozart in Vienna: simple and unassuming he stands there listening, while graceful angels float about, playfully bringing him inspiration from another world. Instead, I must mention the very different type of memorial that was in fact erected to Abel in Christiania and which must greatly disappoint anyone familiar with his nature. On a towering, steep block of granite a youthful athlete of the Byronic type steps over two greyish sacrificial victims, his direction toward the heavens. If needed be, one might take the hero to be a symbol of the human spirit, but one ponders the deeper significance of the two monsters in vain. Are they the conquered quintic equations or elliptic functions? Or the sorrows and cares of his everyday life? The pedestal of the monument bears, in immense letters, the inscription ABEL. == See also == List of things named after Niels Henrik Abel Évariste Galois == Notes == == References == == Further reading == Livio, Mario (2005). The Equation That Couldn't be Solved. New York: Simon & Schuster. ISBN 0-7432-5821-5. Stubhaug, Arild (2000). Niels Henrik Abel and his Times. Trans. by Richard R. Daly. Springer. ISBN 3-540-66834-9. == External links == Biographies and handwritten manuscripts Archived 25 March 2012 at the Wayback Machine from the Abel Prize website O'Connor, John J.; Robertson, Edmund F., "Niels Henrik Abel", MacTutor History of Mathematics Archive, University of St Andrews Biography of Niels Henrik Abel Weisstein, Eric Wolfgang (ed.). "Abel, Niels Henrik (1802–1829)". ScienceWorld. Niels Henrik Abel at the Mathematics Genealogy Project Translation of Niels Henrik Abel's "Research on Elliptic Functions" at Convergence Famous Quotes by Niels Henrik Abel at Convergence Archived 12 February 2006 at the Wayback Machine The Niels Henrik Abel mathematical contest, The Norwegian Mathematical Olympiad Family genealogy
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Wikipedia:Nielsen theory#0
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Nielsen theory is a branch of mathematical research with its origins in topological fixed-point theory. Its central ideas were developed by Danish mathematician Jakob Nielsen, and bear his name. The theory developed in the study of the so-called minimal number of a map f from a compact space to itself, denoted MF[f]. This is defined as: M F [ f ] = min { # F i x ( g ) | g ∼ f } , {\displaystyle {\mathit {MF}}[f]=\min\{\#\mathrm {Fix} (g)\,|\,g\sim f\},} where ~ indicates homotopy of mappings, and #Fix(g) indicates the number of fixed points of g. The minimal number was very difficult to compute in Nielsen's time, and remains so today. Nielsen's approach is to group the fixed-point set into classes, which are judged "essential" or "nonessential" according to whether or not they can be "removed" by a homotopy. Nielsen's original formulation is equivalent to the following: We define an equivalence relation on the set of fixed points of a self-map f on a space X. We say that x is equivalent to y if and only if there exists a path c from x to y with f(c) homotopic to c as paths. The equivalence classes with respect to this relation are called the Nielsen classes of f, and the Nielsen number N(f) is defined as the number of Nielsen classes having non-zero fixed-point index sum. Nielsen proved that N ( f ) ≤ M F [ f ] , {\displaystyle N(f)\leq {\mathit {MF}}[f],} making his invariant a good tool for estimating the much more difficult MF[f]. This leads immediately to what is now known as the Nielsen fixed-point theorem: Any map f has at least N(f) fixed points. Because of its definition in terms of the fixed-point index, the Nielsen number is closely related to the Lefschetz number. Indeed, shortly after Nielsen's initial work, the two invariants were combined into a single "generalized Lefschetz number" (more recently called the Reidemeister trace) by Wecken and Reidemeister. == Bibliography == Fenchel, Werner; Nielsen, Jakob (2003). Asmus L. Schmidt (ed.). Discontinuous groups of isometries in the hyperbolic plane. De Gruyter Studies in mathematics. Vol. 29. Berlin: Walter de Gruyter & Co. == External links == Survey article on Nielsen theory by Robert F. Brown at Topology Atlas
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Wikipedia:Nielsen–Schreier theorem#0
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In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free. It is named after Jakob Nielsen and Otto Schreier. == Statement of the theorem == A free group may be defined from a group presentation consisting of a set of generators with no relations. That is, every element is a product of some sequence of generators and their inverses, but these elements do not obey any equations except those trivially following from gg−1 = 1. The elements of a free group may be described as all possible reduced words, those strings of generators and their inverses in which no generator is adjacent to its own inverse. Two reduced words may be multiplied by concatenating them and then removing any generator-inverse pairs that result from the concatenation. The Nielsen–Schreier theorem states that if H is a subgroup of a free group G, then H is itself isomorphic to a free group. That is, there exists a set S of elements which generate H, with no nontrivial relations among the elements of S. The Nielsen–Schreier formula, or Schreier index formula, quantifies the result in the case where the subgroup has finite index: if G is a free group of rank n (free on n generators), and H is a subgroup of finite index [G : H] = e, then H is free of rank 1 + e ( n − 1 ) {\displaystyle 1+e(n{-}1)} . == Example == Let G be the free group with two generators a , b {\displaystyle a,b} , and let H be the subgroup consisting of all reduced words of even length (products of an even number of letters a , b , a − 1 , b − 1 {\displaystyle a,b,a^{-1},b^{-1}} ). Then H is generated by its six elements p = a a , q = a b , r = b a , s = b b , t = a b − 1 , u = a − 1 b . {\displaystyle p=aa,\ q=ab,\ r=ba,\ s=bb,\ t=ab^{-1},\ u=a^{-1}b.} A factorization of any reduced word in H into these generators and their inverses may be constructed simply by taking consecutive pairs of letters in the reduced word. However, this is not a free presentation of H because the last three generators can be written in terms of the first three as s = r p − 1 q , t = p r − 1 , u = p − 1 q {\displaystyle s=rp^{-1}q,\ t=pr^{-1},\ u=p^{-1}q} . Rather, H is generated as a free group by the three elements p = a a , q = a b , r = b a , {\displaystyle p=aa,\ q=ab,\ r=ba,} which have no relations among them; or instead by several other triples of the six generators. Further, G is free on n = 2 generators, H has index e = [G : H] = 2 in G, and H is free on 1 + e(n–1) = 3 generators. The Nielsen–Schreier theorem states that like H, every subgroup of a free group can be generated as a free group, and if the index of H is finite, its rank is given by the index formula. == Proof == A short proof of the Nielsen–Schreier theorem uses the algebraic topology of fundamental groups and covering spaces. A free group G on a set of generators is the fundamental group of a bouquet of circles, a topological graph X with a single vertex and with a loop-edge for each generator. Any subgroup H of the fundamental group is itself the fundamental group of a connected covering space Y → X. The space Y is a (possibly infinite) topological graph, the Schreier coset graph having one vertex for each coset in G/H. In any connected topological graph, it is possible to shrink the edges of a spanning tree of the graph, producing a bouquet of circles that has the same fundamental group H. Since H is the fundamental group of a bouquet of circles, it is itself free. The rank of H can be computed using two properties of Euler characteristic that follow immediately from its definition. The first property is that the Euler characteristic of a bouquet of s circles is 1 - s. The second property is multiplicativity in covering spaces: If Y is a degree-d cover of X, then χ ( Y ) = d ⋅ χ ( X ) {\displaystyle \chi (Y)=d\cdot \chi (X)} . Now suppose H is a subgroup of the free group G, with index [G:H] = e. The previous part of the proof shows that H is a free group; let r denote the rank of H. Applying the two properties of Euler characteristic for the covering graph Y corresponding to H gives the following: χ ( Y ) = 1 − r {\displaystyle \chi (Y)=1-r} and χ ( Y ) = e ⋅ χ ( X ) = e ( 1 − n ) . {\displaystyle \chi (Y)=e\cdot \chi (X)=e(1-n).} Combining these equations, we obtain r = 1 − e ( 1 − n ) = 1 + e ( n − 1 ) . {\displaystyle r=1-e(1-n)=1+e(n-1).} This proof appears in May's Concise Course. An equivalent proof using homology and the first Betti number of Y is due to Reinhold Baer and Friedrich Levi (1936). The original proof by Schreier forms the Schreier graph in a different way as a quotient of the Cayley graph of G modulo the action of H. According to Schreier's subgroup lemma, a set of generators for a free presentation of H may be constructed from cycles in the covering graph formed by concatenating a spanning tree path from a base point (the coset of the identity) to one of the cosets, a single non-tree edge, and an inverse spanning tree path from the other endpoint of the edge back to the base point. == Axiomatic foundations == Although several different proofs of the Nielsen–Schreier theorem are known, they all depend on the axiom of choice. In the proof based on fundamental groups of bouquets, for instance, the axiom of choice appears in the guise of the statement that every connected graph has a spanning tree. The use of this axiom is necessary, as there exist models of Zermelo–Fraenkel set theory in which the axiom of choice and the Nielsen–Schreier theorem are both false. The Nielsen–Schreier theorem in turn implies a weaker version of the axiom of choice, for finite sets. == History == The Nielsen–Schreier theorem is a non-abelian analogue of an older result of Richard Dedekind, that every subgroup of a free abelian group is free abelian. Jakob Nielsen (1921) originally proved a restricted form of the theorem, stating that any finitely-generated subgroup of a free group is free. His proof involves performing a sequence of Nielsen transformations on the subgroup's generating set that reduce their length (as reduced words in the free group from which they are drawn). Otto Schreier proved the Nielsen–Schreier theorem in its full generality in his 1926 habilitation thesis, Die Untergruppen der freien Gruppe, also published in 1927 in Abh. math. Sem. Hamburg. Univ. The topological proof based on fundamental groups of bouquets of circles is due to Reinhold Baer and Friedrich Levi (1936). Another topological proof, based on the Bass–Serre theory of group actions on trees, was published by Jean-Pierre Serre (1970). == See also == Fundamental theorem of cyclic groups, a similar result for cyclic groups that in the infinite case may be seen as a special case of the Nielsen–Schreier theorem Kurosh subgroup theorem == Notes == == References == Baer, Reinhold; Levi, Friedrich (1936), "Freie Produkte und ihre Untergruppen", Compositio Mathematica, 3: 391–398. Fried, Michael D.; Jarden, Moshe (2008), Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 11 (3rd ed.), Springer-Verlag, p. 70, ISBN 978-3-540-77269-9, Zbl 1145.12001. Howard, Paul E. (1985), "Subgroups of a free group and the axiom of choice", The Journal of Symbolic Logic, 50 (2): 458–467, doi:10.2307/2274234, JSTOR 2274234, MR 0793126, S2CID 33535673. Johnson, D. L. (1980), Topics in the Theory of Group Presentations, London Mathematical Society lecture note series, vol. 42, Cambridge University Press, ISBN 978-0-521-23108-4. Johnson, D. L. (1997), Presentations of Groups, London Mathematical Society student texts, vol. 15 (2nd ed.), Cambridge University Press, ISBN 978-0-521-58542-2. Läuchli, Hans (1962), "Auswahlaxiom in der Algebra", Commentarii Mathematici Helvetici, 37: 1–18, doi:10.1007/bf02566957, hdl:20.500.11850/131689, MR 0143705, S2CID 186223589. Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald (1976), Combinatorial Group Theory (2nd revised ed.), Dover Publications. Nielsen, Jakob (1921), "Om regning med ikke-kommutative faktorer og dens anvendelse i gruppeteorien", Math. Tidsskrift B (in Danish), 1921: 78–94, JFM 48.0123.03. Rotman, Joseph J. (1995), An Introduction to the Theory of Groups, Graduate Texts in Mathematics, vol. 148 (4th ed.), Springer-Verlag, ISBN 978-0-387-94285-8. Serre, J.-P. (1970), Groupes Discretes, Extrait de I'Annuaire du College de France, Paris{{citation}}: CS1 maint: location missing publisher (link). Serre, J.-P. (1980), Trees, Springer-Verlag, ISBN 3-540-10103-9. Stillwell, John (1993), Classical Topology and Combinatorial Group Theory, Graduate Texts in Mathematics, vol. 72 (2nd ed.), Springer-Verlag.
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Wikipedia:Nigel Boston#0
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Nigel Boston (July 20, 1961 – March 31, 2024) was a British-American mathematician, who made notable contributions to algebraic number theory, group theory, and arithmetic geometry. == Biography == Boston attended Harvard University, earning his doctorate in 1987, under supervision of Barry Mazur. He was a Professor Emeritus at the University of Wisconsin–Madison. In 2012, he became a fellow of the American Mathematical Society. Boston died on March 31, 2024, at the age of 62. == References == == External links == Nigel Boston at the Mathematics Genealogy Project
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Wikipedia:Nigel Weiss#0
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Nigel Oscar Weiss FRS (16 December 1936 – 24 June 2020) was an astronomer and mathematician, and leader in the field of astrophysical and geophysical fluid dynamics. He was Emeritus Professor of Mathematical Astrophysics at the University of Cambridge. == Education == Born in South Africa, Weiss studied at Hilton College, Natal, Rugby School and Clare College, Cambridge, and had been a fellow of Clare College since 1965. He read for his PhD in 1961 with a thesis on Variable Hydromagnetic Motions. == Career == In 1987 he became Professor of Mathematical Astrophysics at the University of Cambridge. Between 2000 and 2002 he was President of the Royal Astronomical Society, and in 2007 was awarded the Gold Medal, the society's highest award. == Research == Weiss published extensively in the field of mathematical astrophysics, specialising in solar and stellar magnetic fields, astrophysical and geophysical fluid dynamics and nonlinear dynamical systems. In 1966 he was the first to demonstrate and describe the process of 'flux expulsion' by which a conducting fluid undergoing rotating motion acts to expel the magnetic flux from the region of motion, a process now known to occur in the photosphere of the Sun and other stars. == Awards and honours == Weiss was elected a Fellow of the Royal Society (FRS) in 1992. His nomination reads Professor Weiss is distinguished for his work in the theory of convection, for developing appropriate numerical techniques, and for pioneering their use in precise numerical experiments to gain a qualitative and comprehensive understanding of the behaviour of complicated nonlinear systems. Among many notable achievements in this field, he has been instrumental in the identification of a period-doubling route to chaos in a system of partial differential equations describing doubly-diffusive convection. He has made wide-ranging studies of the magneto-convective processes occurring in the Sun and similar stars. In early work of lasting influence, he analysed the process of magnetic flux expulsion and the mechanism of concentration of magnetic field into ropes from which fluid motion is excluded. In recent work, he has initiated a program of research in the field of nonlinear compressible convection, an important step towards realistic modelling of stellar convection zones. == References ==
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Wikipedia:Nikolai Andreev#0
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Nikolai Nikolayevich Andreev (Russian: Николай Николаевич Андреев) (born 5 February 1975 in Saratov, Russia) is a Russian mathematician and popularizer of mathematics. He was awarded with the Leelavati Award in 2022. == Biography == Nikolai is the Head of the Laboratory for Popularization and Promotion of Mathematics at the Steklov Institute of Mathematics of the Russian Academy of Sciences (Moscow). He received a PhD in mathematics from Moscow State University in 2000. Among his projects is the creation of the website Mathematical Etudes. == Awards and honours == Prize of the President of the Russian Federation in the Area of Sciences and Innovations for Young Scientists (2010) Gold Medal of the Russian Academy of Sciences (2017) for outstanding achievements in science popularization The Leelavati Award in 2022 for his contribution to the art of mathematical animation and mathematical model building, in a style that inspires young and old alike, and that mathematicians around the world can adapt to its many uses, as well as for his tireless efforts to popularize genuine mathematics among the public through videos, lectures, and an award-winning book == References == == External links == Математические этюды [Mathematical Etudes]
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Wikipedia:Nikolai Andreevich Lebedev#0
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Nikolai Andreevich Lebedev (Russian: Никола́й Андре́евич Ле́бедев; August 8, 1919 – January 8, 1982) was a Soviet mathematician who worked on complex function theory and geometric function theory. Jointly with Isaak Milin, he proved the Lebedev–Milin inequalities that were used in the proof of the Bieberbach conjecture. == See also == Conformal map Power series == Biographical references == Aleksandrov, I. A.; Bazilevich, I. E.; Kuz'min, G. V.; Lozinskii, S. M.; Tamrazov, P. M.; Shirokov, N. A. (1983), "Николай Андреевич Лебедев (некролог)", Uspekhi Matematicheskikh Nauk, 38 (2(230)): 195–199, MR 0695474, Zbl 0515.01018, translated in English as "Nikolai Andreevich Lebedev (obituary)", Russian Mathematical Surveys, 38 (2): 177–182, 1983, Bibcode:1983RuMaS..38..177A, doi:10.1070/RM1983v038n02ABEH003472, MR 0695474, Zbl 0528.01020. Goluzina, E. G.; Zhuk, V. V.; Kuzmina, G. V.; Shirokov, N. A. (2001), "Николай Андреевич Лебедев и Ленинградская школа теории функций (50–70 гг.)", Zapiski Nauchnykh Seminarov POMI, 276: 5–19, MR 1850360, Zbl 1077.01525, translated in English as "Nikolai Andreevich Lebedev and the Leningrad School of Function Theory in the 50s–70s", Journal of Mathematical Sciences, 118 (1): 4733–4739, 2003, doi:10.1023/A:1025562313596, MR 1850360, S2CID 117992735, Zbl 1077.01525. == References == Grinshpan, Arcadii Z. (2002), "Logarithmic Geometry, Exponentiation, and Coefficient Bounds in the Theory of Univalent Functions and Nonoverlapping Domains", in Kuhnau, Reiner (ed.), Geometric Function Theory, Handbook of Complex Analysis, vol. 1, Amsterdam: North-Holland, pp. 273–332, ISBN 978-0-444-82845-3, MR 1966197, Zbl 1083.30017. Hayman, W. K. (1994) [1958], Multivalent functions, Cambridge Tracts on Mathematics, vol. 110 (Second ed.), Cambridge: Cambridge University Press, pp. xii+263, ISBN 978-0-521-46026-2, MR 1310776, Zbl 0904.30001. Kuhnau, Reiner, ed. (2002), Geometric Function Theory, Handbook of Complex Analysis, vol. 1, Amsterdam: North-Holland, pp. xii+536, ISBN 978-0-444-82845-3, MR 1966187, Zbl 1057.30001. Milin, I. M. (1977) [1971], Univalent functions and orthonormal systems, Translations of Mathematical Monographs, vol. 49, Providence, R.I.: American Mathematical Society, pp. iv+202, ISBN 978-0-8218-1599-1, MR 0369684, Zbl 0342.30006 (Translation of the 1971 Russian edition, edited by P. L. Duren).
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Wikipedia:Nikolai Ardelyan#0
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Nikolai Vasilievich Ardelyan (Russian: Никола́й Васи́льевич Арделя́н; born 18 September 1953) is a Russian mathematician, Professor, Dr.Sc., Honored Scientist of the Moscow State University, Leading Researcher of the MSU Faculty of Computational Mathematics and Cybernetics. He defended the thesis «Difference-Operational Diagrams of the Dynamics of a Magnetic Gas» by the degree of Doctor of Physical and Mathematical Sciences (1992). Author of 4 books and more than 160 scientific articles. Honored Research Fellow of Moscow State University (2003). == References == == Literature == Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory (1 500 экз ed.). Moscow: Publishing house of Moscow University. 2010. pp. 149–150. ISBN 978-5-211-05838-5 – via Author-compiler Evgeny Grigoriev. == External links == "Nikolai Ardelyan". Department of Computational Methods MSU CMC (in Russian). Retrieved 2018-05-16. "Scientific works of Nikolai Ardelyan". System «TRUE» of the Moscow State University (in Russian). Retrieved 2018-05-16. Math-net.ruScientific works of Sergei Abramov(in English)
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Wikipedia:Nikolai Bakhvalov#0
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Nikolai Sergeevich Bakhvalov (Russian: Николай Серге́евич Бахвалов) (May 29, 1934 – August 29, 2005) was a Soviet and Russian mathematician. Born in Moscow into the family of Sergei Vladimirovich Bakhvalov, a geometer at Moscow State University, N.S. Bakhvalov was exposed to mathematics from a young age. In 1950, Bakhvalov entered the Faculty of Mechanics and Mathematics at Moscow State University. His supervisors there included Kolmogorov and Sobolev. Bakhvalov defended his doctorate in 1958. He was a professor of mathematics at Moscow State University since 1966, specializing in computational mathematics. Bakhvalov was a member of the Russian Academy of Sciences since 1991 and a head of the department of computational mathematics at the college of mechanics and mathematics of the Moscow State University since 1981. Bakhvalov authored over 150 papers, several books, and a popular textbook on numerical methods. He had made major pioneering contributions to many areas of mathematics and mechanics. Starting early in his career, Bakhvalov formulated and proved important results on the optimization of numerical algorithms. In 1959, he determined the complexity of the integration problem in the worst-case setting for integrands of smoothness. Furthermore, he proposed an optimal algorithm for the randomized setting. These can be considered early results in the theory of information-based complexity. Bakhvalov was one of the pioneers of the multigrid method, contributed to the theory of homogenization, and fictitious domain methods. Bakhvalov supervised 47 Ph.D. students and was an advisor to 11 doctorates. == Notes == http://www.netlib.org/na-digest-html/06/v06n04.html#5 Nikolaj S. Bakhvalov: May 29, 1934 - August 29, 2005 From RAS Bakhvalov is sixty == References == Bakhvalov, N.S. (1959), "On the approximate calculation of integrals.", Vestnik MGU, Ser. Mat. Mekh. Astron. Fiz. Khim., 4: 2–18 N. S. Bakhvalov (1966) On the convergence of a relaxation method with natural constraints on the elliptic operator. USSR Comp. Math. Math. Phis.6, 101–13. Homogenisation: Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials (1989), N. S. Bakhvalov, G. Panasenko, Springer, ISBN 978-0-7923-0049-6 N. S. Bakhvalov and A. V. Knyazev (1994) Fictitious domain methods and computation of homogenized properties of composites with a periodic structure of essentially different components, In Numerical Methods and Applications, Ed. Gury I. Marchuk, CRC Press, 221-276. ISBN 978-0-8493-8947-4 Kerimov, M. K. (2006), "Academician Nikolai Sergeevich Bakhvalov (1934--2005)", Computational Mathematics and Mathematical Physics, 46 (1): 182–184, Bibcode:2006CMMPh..46..182K, doi:10.1134/S0965542506010179, S2CID 121126321 == External links == Nikolai Bakhvalov — scientific works on the website Math-Net.Ru Nikolai Bakhvalov at zbMATH Nikolai Bakhvalov at the Mathematics Genealogy Project
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Wikipedia:Nikolai Chebotaryov#0
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Nikolai Grigorievich Chebotaryov (often spelled Chebotarov or Chebotarev; Russian: Никола́й Григо́рьевич Чеботарёв; Ukrainian: Мико́ла Григо́рович Чеботарьо́в; 15 June [O.S. 3 June] 1894 – 2 July 1947) was a Soviet mathematician. He is best known for the Chebotaryov density theorem. He was a student of Dmitry Grave. Chebotaryov worked on the algebra of polynomials, in particular examining the distribution of the zeros. He also studied Galois theory and wrote a textbook on the subject titled Basic Galois Theory. His ideas were used by Emil Artin to prove the Artin reciprocity law. He worked with his student Anatoly Dorodnov on a generalization of the quadrature of the lune, and proved the conjecture now known as the Chebotarev theorem on roots of unity. == Early life == Nikolai Chebotaryov was born on 15 June 1894 in Kamianets-Podilskyi, Russian Empire (now in Ukraine). He entered the department of physics and mathematics at Kiev University in 1912. In 1928, he became a professor at Kazan University, remaining there for the rest of his life. He died on 2 July 1947. He was an atheist. On 14 May 2010, a memorial plaque for Nikolai Chebotaryov was unveiled on the main administration building of I.I. Mechnikov Odessa National University. == References ==
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Wikipedia:Nikolai Günther#0
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Nikolai Maximovich Günther (Russian: Николай Максимович Гюнтер, also transliterated as Nicholas M. Gunther or N. M. Gjunter) (December 17 [O.S. December 5] 1871 – May 4, 1941) was a Russian mathematician known for his work in potential theory and in integral and partial differential equations: later studies have uncovered his contributions to the theory of Gröbner bases. He was an Invited Speaker of the ICM in 1924 at Toronto, in 1928 at Bologna, and in 1932 at Zurich. == Selected publications == Gunther, N. (1932), "Sur les intégrales de Stieltjes et leurs applications aux problèmes de la physique mathématique", Travaux de l'Institute Physico-Mathématique Stekloff (in French), 1: 1–494, JFM 58.1058.01, MR 0031037, Zbl 0006.29703. A large paper aimed at showing the applications of Radon integrals to problems of mathematical physics: the Mathematical Reviews review refers to a 1949 reprint published by the Chelsea Publishing Company. Günther, N. M. (1933), "Sur les opérations linéaires", Physikalische Zeitschrift der Sowjetunion, 3: 115–139, JFM 60.1075.03, Zbl 0008.16601. Gunther, N. M. (1934), La théorie du potentiel et ses applications aux problèmes fondamentaux de la physique mathématique, Collections de monographies sur la théorie des fonctions (in French) (1st ed.), Paris: Gauthier-Villars, p. 303, JFM 60.1127.04, Zbl 0009.11301, reviewed also by Dixon, A. C. (October 1934), "La Théorie du Potentiel et ses applications aux problèmes de la physique mathématique by N. M. Gunther", The Mathematical Gazette, 18 (230): 278, JSTOR 3605383 and by Longley, W. R. (1936), "Review: La Théorie du Potentiel et ses Applications aux Problèmes Fondamentaux de la Physique Mathématique", Bulletin of the American Mathematical Society, 42 (11): 794, doi:10.1090/S0002-9904-1936-06436-0. Günther, N. M. (1967) [1934], Potential theory and its applications to basic problems of mathematical physics, New York: Frederick Ungar Publishing, pp. xi+338, MR 0222316, Zbl 0164.41901. The second edition of the monograph (Gunther 1934), now a classical textbook in potential theory, translated from the Russian original Günther, N. M. (1953) [1934], Теория потенциала и ее применение к основным задачам математической физики (in Russian) (2nd ed.), Москва: Государственное Издательство Технико-Теоретческой Литературы, p. 415, Zbl 0052.10504 (edition cured by V. I. Smirnov and H. L. Smolitskii), which was also translated in German as Günter, N. M. (1957) [1934], Die Potentialtheorie und ihre Anwendung auf Grundaufgaben der mathematischen Physik (in German) (2nd ed.), Leipzig: B. G. Teubner Verlagsgesellschaft, pp. X+314, MR 0109958, Zbl 0077.09702. == See also == Harmonic function Integral equation Radon measure == Notes == == References ==
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Wikipedia:Nikolai Kapustin (mathematician)#0
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Nikolai Yurievich Kapustin (Russian: Никола́й Ю́рьевич Капу́стин; born 3 October 1957) is a Russian mathematician, Professor, Dr. Sc., a professor at the Faculty of Computer Science at the Moscow State University. He defended the thesis "Problems for parabolic-hyperbolic equations and corresponding spectral questions with a parameter at boundary points" for the degree of Doctor of Physical and Mathematical Sciences in 2012 and is the author of three books and more than 90 scientific articles. == References == == Bibliography == Evgeny Grigoriev (2010). Faculty of Computational Mathematics and Cybernetics: History and Modernity: A Biographical Directory (1 500 экз ed.). Moscow: Publishing house of Moscow University. pp. 191–192. ISBN 978-5-211-05838-5. == External links == MSU CMC(in Russian) Scientific works of Nikolai Kapustin Scientific works of Nikolai Kapustin(in English)
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Wikipedia:Nikolai Lobachevsky#0
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Nikolai Ivanovich Lobachevsky (; Russian: Никола́й Ива́нович Лобаче́вский, IPA: [nʲɪkɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕefskʲɪj] ; 1 December [O.S. 20 November] 1792 – 24 February [O.S. 12 February] 1856) was a Russian mathematician and geometer, known primarily for his work on hyperbolic geometry, otherwise known as Lobachevskian geometry, and also for his fundamental study on Dirichlet integrals, known as the Lobachevsky integral formula. William Kingdon Clifford called Lobachevsky the "Copernicus of Geometry" due to the revolutionary character of his work. == Biography == Nikolai Lobachevsky was born either in or near the city of Nizhny Novgorod in the Russian Empire (now in Nizhny Novgorod Oblast, Russia) in 1792 to parents of Russian and Polish origin – Ivan Maksimovich Lobachevsky and Praskovia Alexandrovna Lobachevskaya. He was one of three children. When he was seven, his father, a clerk in a land-surveying office, died, and Nikolai moved with his mother to Kazan. Nikolai Lobachevsky attended Kazan Gymnasium from 1802, graduating in 1807, and then received a scholarship to Kazan University, which had been founded just three years earlier in 1804. At Kazan University, Lobachevsky was influenced by professor Johann Christian Martin Bartels, a former teacher and friend of the German mathematician Carl Friedrich Gauss (1777–1855). Lobachevsky received a Master of Science in physics and mathematics in 1811. In 1814, he became a lecturer at Kazan University, and in 1816, he was promoted to associate professor. In 1822, at the age of 30, he became a full professor, teaching mathematics, physics, and astronomy. He served in many administrative positions and became the rector of Kazan University in 1827. In 1832, he married Varvara Alexeyevna Moiseyeva. They had a large number of children (eighteen according to his son's memoirs, though only seven apparently survived into adulthood). He was dismissed from the university in 1846, ostensibly due to his deteriorating health: by the early 1850s, he was nearly blind and unable to walk. He died in poverty in 1856 and was buried in Arskoe Cemetery, Kazan. In 1811, in his student days, Lobachevsky was accused by a vengeful supervisor of atheism (Russian: признаки безбожия, lit. 'signs of godlessness'). == Career == Lobachevsky's main achievement is the development (independently from János Bolyai) of a non-Euclidean geometry, also referred to as Lobachevskian geometry. Before him, mathematicians were trying to deduce Euclid's fifth postulate from other axioms. Euclid's fifth is a rule in Euclidean geometry which states (in John Playfair's reformulation) that for any given line and point not on the line, there is only one line through the point not intersecting the given line. Lobachevsky would instead develop a geometry in which the fifth postulate was not true. This idea was first reported on 23 [O.S. 1826] February to the session of the department of physics and mathematics, and this research was printed in the periodical 'Kazan University Course Notes' as On the Origin of Geometry (О началах геометрии) between 1829 and 1830. In 1829, Lobachevsky wrote a paper about his ideas called "A Concise Outline of the Foundations of Geometry" that was published by the Kazan Messenger but was rejected when it was submitted to the St. Petersburg Academy of Sciences for publication. The non-Euclidean geometry that Lobachevsky developed is referred to as hyperbolic geometry. Lobachevsky replaced Playfair's axiom with the statement that for any given point there exists more than one line that can be extended through that point and run parallel to another line of which that point is not part. He developed the angle of parallelism which depends on the distance the point is off the given line. In hyperbolic geometry the sum of angles in a hyperbolic triangle must be less than 180 degrees. Non-Euclidean geometry stimulated the development of differential geometry which has many applications. Hyperbolic geometry is frequently referred to as "Lobachevskian geometry" or "Bolyai–Lobachevskian geometry". Some mathematicians and historians have wrongly claimed that Lobachevsky in his studies in non-Euclidean geometry was influenced by Gauss, which is untrue. Gauss himself appreciated Lobachevsky's published works highly, but they never had personal correspondence between them prior to the publication. Although three people—Gauss, Lobachevsky and Bolyai—can be credited with discovery of hyperbolic geometry, Gauss never published his ideas, and Lobachevsky was the first to present his views to the world mathematical community. Lobachevsky's magnum opus Geometriya was completed in 1823, but was not published in its exact original form until 1909, long after he had died. Lobachevsky was also the author of New Foundations of Geometry (1835–1838). He also wrote Geometrical Investigations on the Theory of Parallels (1840) and Pangeometry (1855). Another of Lobachevsky's achievements was developing a method for the approximation of the roots of algebraic equations. This method is now known as the Dandelin–Gräffe method, named after two other mathematicians who discovered it independently. In Russia, it is called the Lobachevsky method. Lobachevsky gave the definition of a function as a correspondence between two sets of real numbers (Peter Gustav Lejeune Dirichlet gave the same definition independently soon after Lobachevsky). == Impact == E. T. Bell wrote about Lobachevsky's influence on the following development of mathematics in his 1937 book Men of Mathematics: The boldness of his challenge and its successful outcome have inspired mathematicians and scientists in general to challenge other "axioms" or accepted "truths", for example the "law" of causality which, for centuries, have seemed as necessary to straight thinking as Euclid's postulate appeared until Lobachevsky discarded it. The full impact of the Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry, for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought. == Honors == 1858 Lobachevskij, an asteroid discovered in 1972, was named in his honour. The lunar crater Lobachevsky was named in his honor. Lobachevsky Prize, a mathematics award by the Kazan State University. The Lobachevsky University was named in his honor. A street in Ploiesti, Romania was named in his honor. == In popular culture == Lobachevsky is the subject of songwriter/mathematician Tom Lehrer's humorous song "Lobachevsky" from his 1953 Songs by Tom Lehrer album. In the song, Lehrer portrays a Russian mathematician who sings about how Lobachevsky influenced him: "And who made me a big success / and brought me wealth and fame? / Nikolai Ivanovich Lobachevsky is his name." Lobachevsky's secret to mathematical success is given as "Plagiarize!", as long as one is always careful to "call it, please, research". According to Lehrer, the song is "not intended as a slur on [Lobachevsky's] character" and the name was chosen "solely for prosodic reasons". The song was based on Danny Kaye and Sylvia Fine's monologue on Stanislavsky and the secret of success in the acting profession. In Poul Anderson's 1969 fantasy novella "Operation Changeling" – which was later expanded into the novel Operation Chaos (1971) – a group of sorcerers navigate a non-Euclidean universe with the assistance of the ghosts of Lobachevsky and Bolyai. Roger Zelazny's science fiction novel Doorways in the Sand contains a poem dedicated to Lobachevsky. In the sitcom 3rd Rock from the Sun, "Dick and the Single Girl" (season 2 episode 24) originally aired on May 11 1997, Sonja Umdahl (Christine Baranski), a forgotten colleague who is transferring to teach at another university, gives as the reason behind her departure that Columbia is the only holder of Nikolai Lobachevsky's manuscripts. == Works == Kagan V. F. (ed.): N. I. Lobachevsky – Complete Collected Works, Vol. I–IV (Russian), Moscow–Leningrad (GITTL), (1946–51). Vol. I: Geometrical Researches on the Theory of Parallels (1840); On the Origin of Geometry (1829–30). Vol. II: New Principles of Geometry with Complete Theory of Parallels (1835–38). Vol. III: Imaginary Geometry (1835); Application of imaginary geometry to certain integrals (1836); Pangeometry (1856). Vol. IV: Works on Other Subjects. English translations Geometrical Researches on the Theory of Parallels. G. B. Halsted (tr.). 1891. Reprinted in Roberto Bonola: Non-Euclidean Geometry: A Critical and Historical Study of its Development. 1912. Dover reprint 1955. Also in: Seth Braver Lobachevski illuminated, MAA 2011. Pangeometry. Excerpts translated by Henry P. Manning: in D. E. Smith A Source Book in Mathematics. McGraw Hill 1929. Dover reprint, pp. 360–374. New Principles of Geometry with Complete Theory of Parallels. G. B. Halsted (tr.). 1897. Nikolai I. Lobachevsky, Pangeometry, translator and editor: A. Papadopoulos, Heritage of European Mathematics Series, Vol. 4, European Mathematical Society. 2010, 310 p. == See also == == References == Notes Citations == External links == O'Connor, John J.; Robertson, Edmund F., "Nikolai Lobachevsky", MacTutor History of Mathematics Archive, University of St Andrews Website dedicated to Lobachevsky (in Spanish) Nikolaj Ivanovič Lobačevskij - Œuvres complètes, tome 2 – Gallica-Math Lobachevsky State University of Nizhny Novgorod
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Wikipedia:Nikolai Shanin#0
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Nikolai Aleksandrovich Shanin (Russian: Николай Александрович Шанин) was a Soviet and Russian mathematician and the founder of a school of constructive mathematics in Leningrad (now Saint Petersburg). He was born on May 25, 1919, in Pskov, Russia, to a family of doctors and passed away on September 17, 2011, in Saint Petersburg, Russia. His father, Alexander Protasyevich Shanin (Russian: Александр Протасьевич Шанин, 1886–1973), was a well-known specialist in skin cancer. In 1935, N. A. Shanin entered the Faculty of Mathematics and Mechanics at Leningrad State University and began his PhD studies there in 1939. Andrey Andreyevich Markov, Jr. became his supervisor, while his second supervisor was Pavel Sergeyevich Alexandrov. Markov’s ideas and personality had a decisive influence on the development of Shanin’s research interests. In 1942, he defended his PhD dissertation, “On the Extension of Topological Spaces,” and in 1946, his D.Sc. dissertation, “On the Product of Topological Spaces.” From 1941 to 1945, during the war between the USSR and Germany, Shanin served in the Red Army. In October 1945, he became a senior research fellow at the Steklov Mathematical Institute of the USSR Academy of Sciences in the Leningrad (later Saint Petersburg) division (LOMI/POMI), where he worked until the end of his life. While working at the Academy of Sciences, he also taught for many years at Leningrad (later Saint Petersburg) State University in the Faculty of Mathematics and Mechanics—where he became a professor in 1957—as well as in the Faculty of Philosophy. Shanin’s research activity can be divided into two periods: topological and logical-constructivist. The first period lasted until the late 1940s. His contributions to general topology remained influential for many years. The second period, which lasted much longer, not only produced numerous scientific results but also led to the formation of a major Leningrad school of mathematical logic and proof theory. This work extended into areas such as computability (e.g., Yuri Matiyasevich), algorithmics, computational complexity, and the application of computers to mathematical research. For A. A. Markov Jr. and later N. A. Shanin, the ineffectiveness of purely existential theorems was a source of "discomfort" in the foundations of mathematics, making the ideas of intuitionism particularly appealing. N. A. Shanin began by generalizing the approach of A. N. Kolmogorov and K. Gödel on embedding operations that transform a formula F of classical logic into a formula Fᶜ' of intuitionistic (constructive) logic, such that Fᶜ' is deducible in intuitionistic logic if and only if F is deducible in classical logic. Moreover, this transformation aimed to preserve the syntax of F as much as possible. Shanin developed a series of sophisticated and general operations and, in particular, described classes of classical formulas containing ∃ and ∨ that remain deducible in intuitionistic logic without modification. This paper was among the first works on intuitionistic logic (a term often replaced by "constructive" logic, in part for political reasons) in the USSR and significantly influenced research in the field. Later, Shanin applied his ideas to other formal systems. N. A. Shanin’s next area of research focused on constructive semantics and was also influenced by intuitionism. However, the semantics of intuitionism was somewhat vague. The first rigorous semantics for intuitionistic logic was S. C. Kleene’s realizability. According to Kleene, a formula ∀x∃y A(x,y) is true if there exists an algorithm that, for each x, constructs y such that A(x,y) holds. In Kleene’s realizability, however, the transformed formula is not necessarily simpler than the original one. Shanin introduced a procedure (algorithm) known as the **elicitation of constructive problems**, which reduces the initial formula to a formula of the form ∃x1...∃xkF , where F contains neither ∃ nor ∨. Due to embedding operations, it then suffices to prove F within classical logic. This procedure significantly facilitated communication between the Russian constructivist school and constructivists in the West, particularly intuitionists. Kleene later observed that, in purely logical terms, Shanin’s algorithm follows from just two principles: Markov's principle and a variant of the Church–Turing thesis. Further development of these ideas led to a **finitary** approach (in the sense of Hilbert) to constructive mathematics. Building on constructive semantics, N. A. Shanin began, in the mid-1950s, a revision of classical mathematics—particularly calculus and functional analysis—from a constructivist perspective. A priori, it is not obvious which notion of a computable real number is the most productive. Shanin defined a **constructive real number** as a "duplex," where both rational approximations and the rate of convergence are given by algorithms, and demonstrated that this approach is effective. Similar algorithmic approaches to real numbers were later developed in the West (see computable number). In 1961, N. A. Shanin organized a **mathematical logic** research group at the Leningrad Department of the Steklov Mathematical Institute of the USSR Academy of Sciences. The group's initial goal was to develop and implement an algorithm for automatic theorem proving, focusing primarily on classical propositional calculus. The first three members of the group were Gennady Davydov (1939–2016), Sergey Maslov (1939–1982), and Grigory Mints (1939–2014). More researchers joined in subsequent years, including V. P. Orevkov [1], A. O. Slissenko, and Yu. V. Matiyasevich. During this period, there was widespread global enthusiasm for automatic theorem proving, particularly in pure logic. Starting from Gentzen’s sequent calculus, Shanin developed a proof search algorithm designed to produce **natural, human-friendly proofs**. He emphasized the use of heuristics and aimed to generate results in the form of **natural deduction**. The algorithm was successfully implemented and demonstrated excellent performance. N. A. Shanin was a dynamic and energetic professor who excelled at explaining fundamental concepts of logic, particularly those lacking formal mathematical definitions, using simpler notions (e.g., integers). His analysis of various semantic issues had a significant influence on philosophers. He had many doctoral students, who work both in Russia and in other countries, including the United States and France. == Further reading == Vsemirnov, M A (2001), "Nikolai Aleksandrovich Shanin (on his 80th birthday)", Russian Math. Surveys, 56 (3): 601–605, Bibcode:2001RuMaS..56..601V, doi:10.1070/RM2001v056n03ABEH000412 Vsemirnov, M A; et al. (2013), "Nikolai Aleksandrovich Shanin (obituary)", Russ. Math. Surv., 68 (4): 763–767, Bibcode:2013RuMaS..68..763V, doi:10.1070/RM2013v068n04ABEH004852 == References == == External links == Nikolai Shanin at the Mathematics Genealogy Project Nikolai Aleksandrovich Shanin at the Steklov Institute of Mathematics at St. Petersburg
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Wikipedia:Nikolas Breuckmann#0
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Nikolas P. Breuckmann (born 1988) is a German mathematical physicist affiliated with the University of Bristol, England. He is, as of Spring 2024, a visiting scientist and program organizer at the Simons Institute for the Theory of Computing at the University of California, Berkeley. His research focuses on quantum information theory, in particular quantum error correction and quantum complexity theory. He is known for his work (together with Anurag Anshu and Chinmay Nirkhe) on proving the NLTS conjecture, a famous open problem in quantum information theory. == Education and early life == Breuckmann was born in Duisburg and grew up in Waltrop, North Rhine-Westphalia, Germany. He earned a BSc in Mathematics and a BSc, an MSc and a PhD in Physics from RWTH Aachen University. His doctoral thesis was titled "Homological Quantum Codes Beyond the Toric Code" and he was supervised by Barbara Terhal. == Career and research == After his PhD, he deferred his University College London Post-Doctoral Fellowship in Quantum Technologies funded by EPSRC for a year to work for Palo Alto-based quantum computing start-up PsiQuantum, which was co-founded by Jeremy O'Brien and Terry Rudolph (among other scientists). In 2022, he became Lecturer (Assistant Professor) in Quantum Computing Theory at the University of Bristol. In 2023, he was awarded the James Clerk Maxwell Medal and Prize by the Institute of Physics for his "outstanding contributions to the quantum error correction field, particularly work on proving the no low-energy trivial state conjecture, a famous open problem in quantum information theory". Quanta Magazine described the proof as "one of the biggest developments in theoretical computer science". This result built on his introduction with Jens Eberhardt of “Balanced Product Quantum Codes”. The NLTS conjecture posits that there exist families of Hamiltonians with all low-energy states of non-trivial complexity. It was formulated in 2013 by Fields Medallist Michael Freedman and Matthew Hastings at Microsoft Research. The conjecture was proven by Breuckmann and colleagues (Anurag Anshu and Chinmay Nirkhe) by showing that the recently discovered families of constant-rate and linear-distance low-density parity-check (LDPC) quantum codes correspond to NLTS local Hamiltonians. This result is a step towards proving the quantum PCP conjecture, considered the most important open problem in quantum complexity theory. He and his former doctoral student Oscar Higgott are inventors of a U.S. patent titled “Subsystem codes with high thresholds by gauge fixing and reduced qubit overhead”, which concerns a technique to significantly improve the performance of quantum error correction in quantum computers. Their related work was included as a major development for computer science in 2023 by Quanta. == References ==
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Wikipedia:Nikolay Korobov#0
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Nikolai Mikhailovich Korobov (Russian: Коробов Николай Михайлович; November 23, 1917 – October 25, 2004) was a Soviet mathematician specializing in number theory and numerical analysis. He is best known for his work in analytic number theory, especially in exponential and trigonometric sums. == References ==
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Wikipedia:Nikolay Krylov (mathematician, born 1941)#0
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Nicolai Vladimirovich Krylov (Russian: Никола́й Влади́мирович Крыло́в; born 5 June 1941) is a Russian mathematician specializing in partial differential equations, particularly stochastic partial differential equations and diffusion processes. Krylov studied at Lomonosov University, where he in 1966 under E. B. Dynkin attained a doctoral candidate title (similar to a PhD) and in 1973 a Russian doctoral degree (somewhat more prestigious than a PhD). He taught from 1966 to 1990 at the Lomonosov University and is since 1990 a professor at the University of Minnesota. At the beginning of his career (starting from 1963) he, in collaboration with Dynkin, worked on nonlinear stochastic control theory, making advances in the study of convex, nonlinear partial equations of 2nd order (i.e. Bellman equations), which were examined with stochastic methods. This led to the Evans-Krylov theory, for which he received with Lawrence C. Evans in 2004 the Leroy P. Steele Prize of the American Mathematical Society (for work done simultaneously and independently by both Krylov and Evans). They proved the second order differentiability (Hölder continuity of the second derivative) of the solutions of convex, completely nonlinear, second order elliptical partial differential equations and thus the existence of "classical solutions" (Theorem of Evans-Krylov). He was in 1978 at Helsinki and in 1986 at Berkeley an invited speaker for the ICM. He received the Humboldt Research Award in 2001. In 1993 he was elected a member of the American Academy of Arts and Sciences (1993). He should not be confused with the mathematician Nikolay M. Krylov. == Works == Controlled diffusion processes, Springer 1980 Introduction to the theory of diffusion processes, AMS 1995 Nonlinear elliptic and parabolic equations of the second order, Dordrecht, Reidel 1987 Lectures on elliptic and parabolic equations in Hölder Spaces, AMS 1996 Introduction to the theory of random processes, AMS 2002 Lectures on Elliptic and Parabolic Equations in Sobolev Spaces, AMS 2008 == References == == External links == Homepage Nikolay Krylov at the Mathematics Genealogy Project
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Wikipedia:Nikolay Morozkin#0
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Nikolai Danilovich Morozkin (Russian: Николай Данилович Морозкин; born 27 November 1953) is a Soviet and Russian mathematician. Doctor of Physical and Mathematical Sciences (1996), Professor (1997), Rector of Bashkir State University (since 2013), Honorary Figure of Higher Education of the Russian Federation (2011). == Life == Nikolai Morozkin graduated from Mordovia State University in 1975. In 1980 he finished his postgraduate studies at the Faculty of Applied Mathematics in Leningrad State University. In 1980–1983 he worked as a senior lecturer for the Department of Differential Equations in Mordovia State University. In 1983 he started to teach at Bashkir State University, first as a senior lecturer, docent. In 1993 he became the Head of the Department of Applied Informatics and Numerical Methods, and since 1997 he simultaneously worked as a vice-rector for academic affairs. On May 16, 2021, Nikolai Morozkin was elected Rector of Bashkir State University. In October 2017, he was elected again for the next period. Nikolai Morozkin is a member of The Board of Trustees of Bashkir Regional Branch of The Russian Geographical Society. == Research == In 1980, he defended his Candidate's dissertation; in 1996 – his Doctorate dissertation. In 1997, he became a professor. Areas of research: Mathematical Modelling, Informatics, Control Optimization, Numerical Methods Development. Nikolai Morozkin created problem-solving procedures for speed-in-action problems with non-linear convex phase constraints and for heat optimizations problems accounting for various technological constraints and strength properties of heated objects. He conducted research in mathematical modeling and calculation of pressure field in medical and biological systems, for example, in в temporomandibular joint. Nikolai Morozkin is a member of Doctorate Dissertation Defence Boards for the following specialties: thermophysics and thermology; fluid, gas, and plasma mechanics. He is an author of about 100 scientific papers, 2 utility patents. == Selected works == === In Russian === О сходимости конечномерных приближений в задаче оптимального одномерного нагрева с учетом фазовых ограничений // Журнал вычислительной математики и математической физики. — 1996. — No. 10. — С. 12–22. Об одном итерационном методе решения задачи оптимального нелинейного нагрева с фазовыми ограничениями // Журнал вычислительной математики и математической физики. — 2000. — Т. 40, No. 11. — С. 1615—1632 (соавторы Голичев И. И., Дульцев А. В.). Оптимальное управление одномерным нагревом с учетом фазовых ограничений // Математическое моделирование. — 1996. — Т. 8, No. 3. — С. 91–110. Оптимальное управление процессами нагрева с учетом фазовых ограничений: Учебное пособие. — Уфа: изд-во БашГУ, 1997. — 114 с. Оптимизация высокотемпературного индукционного нагрева сплошного цилиндра с учетом ограничений на термонапряжения // Электричество. — 1995. — No. 5. — С. 56–60. Применение метода конечных элементов для расчета пластового давления в нефтяном месторождении // Нефтяное хозяйство. — 1998. — No. 11. — С. 28—30 (соавтор Бикбулатова Г. С.). Расчет поля напряжений нижнечелюстного сустава // Математическое моделирование. — 2008. — Т. 20, No. 6. — С. 119–128. (соавтор Колонских Д. М.). === In English === The Convergence of Finite — Dimensional Approximations in the Problem of the Optimal One — Dimensional Heating Taking Phase Constraints into Account // Comp. Math. Phys. — 1996. — Vol. 36, N 10. — P. 1331–1339. == Titles and awards == Honored Worker of Higher Education of the Russian Federation (2002) Honored Worker of Science of the Republic of Bashkortostan (2004) Honorary Figure of Russian Higher Education == References ==
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Wikipedia:Nikolay Nekhoroshev#0
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Nikolai Nikolaevich Nekhoroshev (Russian: Николай Николаевич Нехорошев; 2 October 1946 – 18 October 2008) was a prominent Soviet Russian mathematician specializing in classical mechanics and dynamical systems. His research concerned Hamiltonian mechanics, perturbation theory, celestial mechanics, integrable systems, dynamical systems, the quasiclassical approximation, and singularity theory. He proved, in particular, a stability result in KAM-theory stating that, under certain conditions, solutions of nearly integrable systems stay close to invariant tori for exponentially long times (Giorgilli 1989). Nekhoroshev was professor of the Moscow State University and University of Milan. He was an alumnus of Moscow's boarding school no. 18 (1964). Winner of the Kolmogorov Prize (1997). == References == Nekhoroshev, N.N. (1977), "On the behavior of near integrable systems", in Anosov, D.V. (ed.), 20 Lectures Delivered at the International Congress of Mathematicians in Vancouver, 1974, AMS Translations 2, vol. 109, N.Y.: AMS Bookstore, pp. 87–90, ISBN 978-0-821-83059-8. Giorgilli, Antonio (1989), "Effective stability in Hamiltonian systems in the light of Nekhoroshev's theorem", Integrable Systems and Applications, Lecture Notes in Physics, vol. 342, Berlin, New York: Springer-Verlag, pp. 142–153, doi:10.1007/BFb0035669, ISBN 978-3-540-51615-6. Abramov, A.M.; Arnold, Vl.I.; et al. (2009), "Nikolai Nikolaevich Nekhoroshev (obituary)", Uspekhi Mat. Nauk (in Russian), 64 (3(387)): 174–178, Bibcode:2009RuMaS..64..561A, doi:10.1070/RM2009v064n03ABEH004622.
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Wikipedia:Nikolay Yakovlevich Sonin#0
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Nikolay Yakovlevich Sonin (Russian: Никола́й Я́ковлевич Со́нин, February 22, 1849 – February 27, 1915) was a Russian mathematician. == Biography == He was born in Tula and attended Lomonosov University, studying mathematics and physics there from 1865 to 1869. His advisor was Nikolai Bugaev. He obtained a master's degree with a thesis submitted in 1871, then he taught at the University of Warsaw where he obtained a doctorate in 1874. He was appointed to a chair in the University of Warsaw in 1876. In 1894, Sonin moved to St. Petersburg, where he taught at the University for Women. Sonin worked on special functions, in particular cylindrical functions. For instance, the Sonine formula is a formula given by Sonin for the integral of the product of three Bessel functions. He is furthermore credited with the introduction of the associated Laguerre polynomials. He also contributed to the Euler–Maclaurin summation formula. Other topics Sonin studied include Bernoulli polynomials and approximate computation of definite integrals, continuing Chebyshev's work on numerical integration. Together with Andrey Markov, Sonin prepared a two volume edition of Chebyshev's works in French and Russian. He died in St. Petersburg. == References == == External links == Nikolay Yakovlevich Sonin at the Mathematics Genealogy Project
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Wikipedia:Nilakantha Somayaji#0
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Keļallur Nīlakaṇṭha Somayāji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501. He had also composed an elaborate commentary on Aryabhatiya called the Aryabhatiya Bhasya. In this Bhasya, Nilakantha had discussed infinite series expansions of trigonometric functions and problems of algebra and spherical geometry. Grahapariksakrama is a manual on making observations in astronomy based on instruments of the time. == Early life == Nilakantha was born into a Brahmin family which came from South Malabar in Kerala. == Biographical details == Nilakantha Somayaji was one of the very few authors of the scholarly traditions of India who had cared to record details about his own life and times. In one of his works titled Siddhanta-star and also in his own commentary on Siddhanta-darpana, Nilakantha Somayaji stated that he was born on Kali-day 1,660,181 which works out to 14 June 1444 CE. A contemporary reference to Nilakantha Somayaji in a Malayalam work on astrology implies that Somayaji lived to a ripe old age even to become a centenarian. Sankara Variar, a pupil of Nilakantha Somayaji, in his commentary on Tantrasamgraha titled Tantrasamgraha-vyakhya, points out that the first and last verses of Tantrasamgraha contain chronograms specifying the Kali-days of the commencement (1,680,548) and of completion (1,680,553) of Somayaji's magnum opus Tantrasamgraha. Both these days occur in 1500 CE. In Aryabhatiya-bhashya, Nilakantha Somayaji has stated that he was the son of Jatavedas and he had a brother named Sankara. Somayaji has further stated that he was a Bhatta belonging to the Gargya gotra and was a follower of Asvalayana-sutra of Rigveda. References in his own Laghuramayana indicate that Nilakantha Somayaji was a member of the Kelallur family (Sanskritised as Kerala-sad-grama) residing at Kundagrama, now known as Trikkandiyur in modern Tirur, Kerala. His wife was named Arya and he had two sons Rama and Dakshinamurti. Nilakantha Somayaji studied vedanta and some aspects of astronomy under one Ravi. However, it was Damodara, son of Kerala-drgganita author Paramesvara, who initiated him into the science of astronomy and instructed him in the basic principles of mathematical computations. The great Malayalam poet Thunchaththu Ramanujan Ezhuthachan is said to have been a student of Nilakantha Somayaji. The epithet Somayaji is a title assigned to or assumed by a Namputiri who has performed the vedic ritual of Somayajna. So it could be surmised that Nilakantha Somayaji had also performed a Somayajna ritual and assumed the title of a Somayaji in later life. In colloquial Malayalam usage the word Somayaji has been corrupted to Comatiri. Somayāji’s "Āryabhaṭīyabhāṣya" is the most extensive commentary on Āryabhaṭīya. He takes all pains to expose the rationale and the objective behind the statements and observations made by Āryabhaṭa. == Nilakantha Somayaji as a polymath == Nilakantha's writings substantiate his knowledge of several branches of Indian philosophy and culture. It is said that he could refer to a Mimamsa authority to establish his view-point in a debate and with equal felicity apply a grammatical dictum to the same purpose. In his writings he refers to a Mimamsa authority, quotes extensively from Pingala's chandas-sutra, scriptures, Dharmasastras, Bhagavata and Vishnupurana also. Sundararaja, a contemporary Tamil astronomer, refers to Nilakantha as sad-darshani-parangata, one who had mastered the six systems of Indian philosophy. == Astronomy == In his Tantrasangraha, Nilakantha revised Aryabhata's model for the planets Mercury and Venus. According to George G. Joseph his equation of the centre for these planets remained the most accurate until the time of Johannes Kepler in the 17th century. In his Aryabhatiyabhasya, a commentary on Aryabhata's Aryabhatiya, Nilakantha developed a computational system for a partially heliocentric planetary model in which Mercury, Venus, Mars, Jupiter and Saturn orbit the Sun, which in turn orbits the Earth, similar to the Tychonic system later proposed by Tycho Brahe in the late 16th century. Most astronomers of the Kerala school who followed him accepted this planetary model. Somayaji time and again advocates the necessity of periodical modification of computation system based on observations and experimentations. One of his works, Jyotirmimamsa, is written exclusively for this purpose. His critical and analytical approach is reflected in all works. == Works == The following is a brief description of the works by Nilakantha Somayaji dealing with astronomy and mathematics. Tantrasamgraha Golasara : Description of basic astronomical elements and procedures Sidhhantadarpana : A short work in 32 slokas enunciating the astronomical constants with reference to the Kalpa and specifying his views on astronomical concepts and topics. Candrachayaganita : A work in 32 verses on the methods for the calculation of time from the measurement of the shadow of the gnomon cast by the moon and vice versa. Aryabhatiya-bhashya : Elaborate commentary on Aryabhatiya. Sidhhantadarpana-vyakhya : Commentary on his own Siddhantadarapana. Chandrachhayaganita-vyakhya : Commentary on his own Chandrachhayaganita. Sundaraja-prasnottara : Nilakantha's answers to questions posed by Sundaraja, a Tamil Nadu-based astronomer. Grahanadi-grantha : Rationale of the necessity of correcting old astronomical constants by observations. Grahapariksakrama : Description of the principles and methods for verifying astronomical computations by regular observations. Jyotirmimamsa : Analysis of astronomy == See also == Indian mathematicians Indian astronomy List of astronomers and mathematicians of the Kerala school == References == == Further reading == N. K. Sundareswaran (2009). The Contribution of Kelallur Nilakantha Somayaji to Astronomy (2009 ed.). Calicut: University of Calicut. ISBN 9788177481495. R.C. Gupta. "Second order interpolation in Indian mathematics up to the fifteenth century" (PDF). Indian Journal of History of Science. 4 (1 & 2): 87–98. Archived from the original (PDF) on 9 March 2012. K. V. Sarma (2008) "Nilakantha Somayaji", Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures (2nd edition) edited by Helaine Selin, Springer, ISBN 978-1-4020-4559-2. Shailesh A Shirali (May 1997). "Nilakantha, Euler and pi". Resonance: 28–43. Retrieved 6 September 2016. Ranjan Roy (December 1990). "The discovery of the series formula for π by Leibnitz, Gregory and Nilakantha". Mathematics Magazine. 63 (5). Mathematical Association of America: 291–306. doi:10.2307/2690896. JSTOR 2690896. Retrieved 6 September 2016. == External links == Official Website of Chenthala Vishnu Temple Tantrasamgraha Official Website Official Website of Kelallur Nilakantha Somayaji (archived)
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Wikipedia:Nilima Nigam#0
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Nilima Nigam is an Indian and Canadian applied mathematician specializing in numerical analysis, partial differential equations, and mathematical models, particularly in problems of mathematical physiology involving muscular, skeletal, and cancer tissue in human bodies. She is a professor of mathematics at Simon Fraser University. == Education and career == Nigam was a physics student at IIT Kharagpur in India, where she graduated with honours in 1994. She went to the University of Delaware for graduate study in mathematics, earned a master's degree there in 1996, and completed her Ph.D. in 1999. Her dissertation, Variational Methods for a Class of Boundary Value Problems Exterior to a Thin Domain, was supervised by George Chia-Chu Hsiao. After postdoctoral research at the Institute for Mathematics and its Applications at the University of Minnesota, she moved to Canada as an assistant professor of mathematics and statistics at McGill University, in 2001. McGill gave her tenure as an associate professor in 2008, the same year that she moved to her present position at Simon Fraser University. From 2008 to 2010, she was also associate scientific director at Mitacs, a Canadian nonprofit research organization, and from 2008 to 2014 she held a tier II Canada Research Chair in applied mathematics. She was promoted to full professor at Simon Fraser in 2013. == Recognition == Nigam was elected as a Fellow of the Canadian Mathematical Society in 2023. == References == == External links == Home page
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Wikipedia:Nilpotent cone#0
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In mathematics, the nilpotent cone N {\displaystyle {\mathcal {N}}} of a finite-dimensional semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} is the set of elements that act nilpotently in all representations of g . {\displaystyle {\mathfrak {g}}.} In other words, N = { a ∈ g : ρ ( a ) is nilpotent for all representations ρ : g → End ( V ) } . {\displaystyle {\mathcal {N}}=\{a\in {\mathfrak {g}}:\rho (a){\mbox{ is nilpotent for all representations }}\rho :{\mathfrak {g}}\to \operatorname {End} (V)\}.} The nilpotent cone is an irreducible subvariety of g {\displaystyle {\mathfrak {g}}} (considered as a vector space). == Example == The nilpotent cone of sl 2 {\displaystyle \operatorname {sl} _{2}} , the Lie algebra of 2×2 matrices with vanishing trace, is the variety of all 2×2 traceless matrices with rank less than or equal to 1. {\displaystyle 1.} == References == Aoki, T.; Majima, H.; Takei, Y.; Tose, N. (2009), Algebraic Analysis of Differential Equations: from Microlocal Analysis to Exponential Asymptotics, Springer, p. 173, ISBN 9784431732402. Anker, Jean-Philippe; Orsted, Bent (2006), Lie Theory: Unitary Representations and Compactifications of Symmetric Spaces, Progress in Mathematics, vol. 229, Birkhäuser, p. 166, ISBN 9780817644307. This article incorporates material from Nilpotent cone on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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Wikipedia:Nilüfer Çınar Çorlulu#0
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Nilüfer Çinar Çorlulu (born Nilüfer Çınar on November 4, 1962) is a Turkish Woman International Master (WIM) of chess. With nine national champion titles, she is one of the most successful female chess players in Turkey, being only second after Gülümser Öney, who has eleven titles, and equaled in 2013 by Betül Cemre Yıldız. == Early years == Nilüfer was born in 1962 in İskenderun, southern Turkey. In high school, she was a successful basketball player, and her teacher recommended that she attend sports academy after graduation. She got acquainted with chess quite late through a friend during her first year at the Black Sea Technical University in Trabzon, where she studied mathematics. She soon realized that her friend did not even know the rules of chess well. The more she learned chess, the more her lifestyle changed. == Chess career == In 1984, she participated in a tournament organized by the Student Affairs Directoriate held in İnciraltı, İzmir and took third place. The same year, Nilüfer took part at the Turkish Chess Championship and placed sixth. Getting more ambitious, she thought even about dropping out of university in order to devote herself to chess playing completely. In 1986, she won second place at the national championship. Nilüfer took part at the 27th Chess Olympiad in Dubai, United Arab Emirates in 1986, reaching in total 9½/14. In 1987 she became the Turkish champion, and repeated her success for six consecutive years until 1992. After her second place at the zonal tournament in Greece in 1993, she was awarded the title of Woman International Master by FIDE, as the first ever Turkish woman receiving this honor. She has complained about the lack of an expert trainer in that time due to the Turkish Chess Federation's financial incapability to support her. She decided then to pause a while because there was no higher goal for her. Still in the national team, she returned to chess in 1999, and became again national champion, defending her title in two following years. She has admitted that in the 1980s and 1990s women's chess in Turkey was not at a high level. Nilüfer was a member of the Turkish national team for 21 years. A graduate of mathematics, she serves as a trainer and consultant of chess, mathematics and geometry as well as mental arithmetic for children aged 5–14 in her own firm based in Ankara. Nilüfer Çınar Çorlulu plays also simultaneous chess with school children to get them interested in chess. She serves as the Turkish Chess Federation's province representative in Ankara. == Achievements == Turkish Chess Championship 1986 – 2nd place 1987–1992 (six times), 1999–2001 (three times) – champion == References == == External links == Nilufer Cinar Corlulu rating card at FIDE Nilufer Cinar Corlulu player profile and games at Chessgames.com Nilüfer Çınar Çorlulu Women's Chess Olympiad record at OlimpBase.org
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Wikipedia:Nina Bari#0
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Nina Karlovna Bari (Russian: Нина Карловна Бари; 19 November 1901 – 15 July 1961) was a Soviet mathematician known for her work on trigonometric series. She is also well-known for two textbooks, Higher Algebra and The Theory of Series. == Early life and education == Nina Bari was born in Russia on 19 November 1901, the daughter of Olga and Karl Adolfovich Bari, a physician. In 1918, she became one of the first women to be accepted to the Department of Physics and Mathematics at the prestigious Moscow State University. She graduated in 1921—just three years after entering the university. After graduation, Bari began her teaching career. She lectured at the Moscow Forestry Institute, the Moscow Polytechnic Institute, and the Sverdlov Communist Institute. Bari applied for and received the only paid research fellowship awarded by the newly created Research Institute of Mathematics and Mechanics. As a student, Bari was drawn to an elite group nicknamed the Luzitania—an informal academic and social organization. She studied trigonometric series and functions under the tutelage of Nikolai Luzin, becoming one of his star students. She presented the main result of her research to the Moscow Mathematical Society in 1922—the first woman to address the society. In 1926, Bari completed her doctoral work on the topic of trigonometric expansions, winning the Glavnauk Prize for her thesis work. In 1927, Bari took advantage of an opportunity to study in Paris at the Sorbonne and the College de France. She then attended the Polish Mathematical Congress in Lwów, Poland; a Rockefeller grant enabled her to return to Paris to continue her studies. Bari's decision to travel may have been influenced by the disintegration of the Luzitanians. Luzin's irascible, demanding personality had alienated many of the mathematicians who had gathered around him. By 1930, all traces of the Luzitania movement had vanished, and Luzin left Moscow State for the Academy of Science's Steklov Institute of Mathematics. In 1932, she became a professor at Moscow State University and in 1935 was awarded the title of Doctor of Physical and Mathematical Sciences, a more prestigious research degree than traditional Ph.D. By this time, she had completed foundational work on trigonometric series. == Career and later life == She was a close collaborator with Dmitrii Menshov on a number of research projects. She and Menshov took charge of function theory work at Moscow State during the 1940s. In 1952, she published an important piece on primitive functions, and trigonometric series and their almost everywhere convergence. Bari also posted works at the 1956 Third All-Union Congress in Moscow and the 1958 International Congress of Mathematicians in Edinburgh. Mathematics was the center of Bari's intellectual life, but she enjoyed literature and the arts. She was also a mountain hiking enthusiast and tackled the Caucasus, Altai, Pamir and Tian Shan mountain ranges in Russia. Bari's interest in mountain hiking was inspired by her husband, Viktor Vladimirovich Nemytskii, a Soviet mathematician, Moscow State professor and an avid mountain explorer. There is no documentation of their marriage available, but contemporaries believe the two married later in life. Bari's last work—her 55th publication—was a 900-page monograph on the state of the art of trigonometric series theory, which is recognized as a standard reference work for those specializing in function and trigonometric series theory. == Death == On 15 July 1961, Bari died after being hit by a train. It was possibly a suicide due to depression caused by Luzin's death eleven years earlier. == References ==
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Wikipedia:Nina Holden#0
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Nina Holden is a Norwegian mathematician interested in probability theory and stochastic processes, including graphons, random planar maps, the Schramm–Loewner evolution, and their applications to quantum gravity. She was a Junior Fellow at the Institute for Theoretical Studies at ETH Zurich, and is currently an associate professor at the Courant Institute of Mathematical Sciences of New York University. == Education == As a student at Berg Upper Secondary School in Oslo, Norway, Holden became the first woman to win the Abel competition, Norway's national Mathematical Olympiad. She competed in 2005 in the International Mathematical Olympiad, where she earned an honorable mention with one of the two top scores on the Norwegian team. She became a student at the University of Oslo in Norway, where she earned a bachelor's degree in mathematics and computational science in 2008 and a master's degree in applied mathematics in 2010. While a student in Oslo, she also visited the University of Oxford from 2006 to 2007. After three years of work as an energy market analyst, she went to the Massachusetts Institute of Technology for graduate study, and completed her Ph.D. there in 2018. Her dissertation, Cardy embedding of random planar maps and a KPZ formula for mated trees, was supervised by Scott Sheffield. == Recognition == In association with the 2021 Breakthrough Prizes, Holden was awarded one of three 2021 Maryam Mirzakhani New Frontiers Prizes, for early-career achievements by a woman mathematician. The citation reads: "for work in random geometry, particularly on Liouville Quantum Gravity as a scaling limit of random triangulations." The particular work refers to her joint work with Xin Sun on the convergence of uniform triangulations under a conformal embedding. The other two winners of the prize were Urmila Mahadev and Lisa Piccirillo. In 2023, she was a recipient of the Rollo Davidson Prize, and in the following year, she was awarded the EMS Prize "for her profound contributions to probability theory and its applications to statistical physics, including results linking Liouville quantum gravity, the Schramm-Loewner evolution, and random triangulations". == References == == External links == Home page at CIMS NYU
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Wikipedia:Nina Snaith#0
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Nina Claire Snaith is a British mathematician at the University of Bristol working in random matrix theory and quantum chaos. == Education == Snaith was educated at the University of Bristol where she received her PhD in 2000 for research supervised by Jonathan Keating. == Career and research == In 1998, Snaith and her then adviser Jonathan Keating conjectured a value for the leading coefficient of the asymptotics of the moments of the Riemann zeta function. Keating and Snaith's guessed value for the constant was based on random-matrix theory, following a trend that started with Montgomery's pair correlation conjecture. Keating's and Snaith's work extended works by Brian Conrey, Ghosh, and Gonek, also conjectural, based on number theoretic heuristics; Conrey, Farmer, Keating, Rubinstein, and Snaith later conjectured the lower terms in the asymptotics of the moments. Snaith's work appeared in her doctoral thesis Random Matrix Theory and zeta functions. Snaith is currently Professor of Mathematical Physics at the University of Bristol. === Awards and honours === In 2008, Snaith was awarded the London Mathematical Society's Whitehead Prize. In 2014, she delivered the annual Hanna Neumann Lecture to honour the achievements of women in mathematics. == Personal life == Snaith is the daughter of mathematician Victor Snaith and sister of mathematician and musician Dan Snaith, mostly known by his artistic names Manitoba, Caribou, and Daphni. == References ==
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Wikipedia:Nina Uraltseva#0
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Nina Nikolaevna Uraltseva (born 1934, Russian: Нина Николаевна Уральцева) is a Russian mathematician, a professor of mathematics and head of the department of mathematical physics at Saint Petersburg State University, and the editor-in-chief of the Proceedings of the St. Petersburg Mathematical Society. Her specialty is the study of nonlinear partial differential equations. Nina Uraltseva was born on 24 May 1934 in Leningrad, USSR (currently St. Petersburg, Russia), to parents Nikolai Fedorovich Uraltsev, (an engineer) and Lidiya Ivanovna Zmanovskaya (a school physics teacher). She received a diploma in physics from Saint Petersburg State University (then known as Leningrad State University) in 1956. She earned a Ph.D. in 1960 from the same university, under the supervision of Olga Ladyzhenskaya, and completed her D.Sc. (the Soviet equivalent of a habilitation) in 1964. She joined the faculty of Leningrad State University in 1959, and was promoted to professor in 1968 and department head in 1974. Uraltseva's research on Hilbert's nineteenth problem and Hilbert's twentieth problem led to her recognition in 1967 by the Chebyshev Prize of the USSR Academy of Sciences, and in 1969 by the USSR State Prize. Uraltseva was a speaker at the 1970 and 1986 International Congress of Mathematicians. A meeting on partial differential equations was held in honor of her 70th birthday at the Royal Institute of Technology, Stockholm, in June 2005, and in 2006 the Royal Institute of Technology gave her an honorary doctorate. For her 75th birthday, a book on partial differential equations and a special issue of the journal Problemy Matematicheskogo Analiza were dedicated to her. A special issue of the journal Algebra i Analiz commemorated her 85th birthday. == References ==
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Wikipedia:Nine-point stencil#0
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In numerical analysis, given a square grid in two dimensions, the nine-point stencil of a point in the grid is a stencil made up of the point itself together with its eight "neighbors". It is used to write finite difference approximations to derivatives at grid points. It is an example for numerical differentiation. This stencil is often used to approximate the Laplacian of a function of two variables. == Motivation == If we discretize the 2D Laplacian by using central-difference methods, we obtain the commonly used five-point stencil, represented by the following convolution kernel: D C D = [ 0 1 0 1 − 4 1 0 1 0 ] {\displaystyle D_{CD}={\begin{bmatrix}0&1&0\\1&-4&1\\0&1&0\end{bmatrix}}} Even though it is simple to obtain and computationally lighter, the central difference kernel possess an undesired intrinsic anisotropic property, since it doesn't take into account the diagonal neighbours. This intrinsic anisotropy poses a problem when applied on certain numerical simulations or when more accuracy is required, by propagating the Laplacian effect faster in the coordinate axes directions and slower in the other directions, thus distorting the final result. This drawback calls for finding better methods for discretizing the Laplacian, reducing or eliminating the anisotropy. == Implementation == The two most commonly used isotropic nine-point stencils are displayed below, in their convolution kernel forms. They can be obtained by the following formula: D = ( 1 − γ ) [ 0 1 0 1 − 4 1 0 1 0 ] + γ [ 1 / 2 0 1 / 2 0 − 2 0 1 / 2 0 1 / 2 ] {\displaystyle D=(1-\gamma ){\begin{bmatrix}0&1&0\\1&-4&1\\0&1&0\end{bmatrix}}+\gamma {\begin{bmatrix}1/2&0&1/2\\0&-2&0\\1/2&0&1/2\end{bmatrix}}} The first one is known by Oono-Puri, and it is obtained when γ=1/2. D O P = [ 1 / 4 2 / 4 1 / 4 2 / 4 − 12 / 4 2 / 4 1 / 4 2 / 4 1 / 4 ] = [ 0.25 0.5 0.25 0.5 − 3 0.5 0.25 0.5 0.25 ] {\displaystyle D_{OP}={\begin{bmatrix}1/4&2/4&1/4\\2/4&-12/4&2/4\\1/4&2/4&1/4\end{bmatrix}}={\begin{bmatrix}0.25&0.5&0.25\\0.5&-3&0.5\\0.25&0.5&0.25\end{bmatrix}}} The second one is known by Patra-Karttunen or Mehrstellen, and it is obtained when γ=1/3. D P K = [ 1 / 6 4 / 6 1 / 6 4 / 6 − 20 / 6 4 / 6 1 / 6 4 / 6 1 / 6 ] = [ 0.16 0.66 0.16 0.66 − 3.33 0.66 0.16 0.66 0.16 ] {\displaystyle D_{PK}={\begin{bmatrix}1/6&4/6&1/6\\4/6&-20/6&4/6\\1/6&4/6&1/6\end{bmatrix}}={\begin{bmatrix}0.16&0.66&0.16\\0.66&-3.33&0.66\\0.16&0.66&0.16\end{bmatrix}}} Both are isotropic forms of discrete Laplacian, and in the limit of small Δx, they all become equivalent, as Oono-Puri being described as the optimally isotropic form of discretization, displaying reduced overall error, and Patra-Karttunen having been systematically derived by imposing conditions of rotational invariance, displaying smallest error around the origin. == Desired anisotropy == On the other hand, if controlled anisotropic effects are a desired feature, when solving anisotropic diffusion problems for example, it is also possible to use the 9-point stencil combined with tensors to generate them. Consider the laplacian in the following form: c ∇ 2 A = c D O P ∗ A {\displaystyle c\nabla ^{2}A=cD_{OP}*A} Where c is just a constant coefficient. Now if we replace c by the 2nd rank tensor C: C = [ c 1 0 0 c 2 ] {\displaystyle C={\begin{bmatrix}c_{1}&0\\0&c_{2}\end{bmatrix}}} Where c1 is the constant coefficient for the principal direction in x axis, and c2 is the constant coefficient for the secondary direction in y axis. In order to generate anisotropic effects, c1 and c2 must be different. By multiplying it by the rotation matrix Q, we obtain C', allowing anisotropic propagations in arbitrary directions other than the coordinate axes. Q = [ cos θ sin θ − sin θ cos θ ] {\displaystyle Q={\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}} C ′ = Q C Q T {\displaystyle C'=QCQ^{\operatorname {T} }} C ′ = [ c x x c x y c x y c y y ] = [ c 1 cos 2 θ + c 2 sin 2 θ ( c 2 − c 1 ) cos θ sin θ ( c 2 − c 1 ) cos θ sin θ c 2 cos 2 θ + c 1 sin 2 θ ] {\displaystyle C'={\begin{bmatrix}c_{xx}&c_{xy}\\c_{xy}&c_{yy}\end{bmatrix}}={\begin{bmatrix}c_{1}\cos ^{2}\theta +c_{2}\sin ^{2}\theta &(c_{2}-c_{1})\cos \theta \sin \theta \\(c_{2}-c_{1})\cos \theta \sin \theta &c_{2}\cos ^{2}\theta +c_{1}\sin ^{2}\theta \end{bmatrix}}} Which is very similar to the Cauchy stress tensor in 2 dimensions. The angle θ {\displaystyle \theta } can be obtained by generating a vector field V = V x i + V y j {\displaystyle \mathbf {V} =V_{x}{\mathbf {i} }+V_{y}{\mathbf {j} }} in order to orientate the pattern as desired. Then: θ = arctan ( V y / V x ) {\displaystyle \theta =\arctan(V_{y}/V_{x})} Or, for different anisotropic effects using the same vector field θ = arctan ( V y / − V x ) {\displaystyle \theta =\arctan(V_{y}/-V_{x})} It is important to note that, regardless of the values of θ {\displaystyle \theta } , the anisotropic propagation will occur parallel to the secondary direction c2 and perpendicular to the principal direction c1:. The resulting convolution kernel is as follows D A n i s o = [ − c x y 2 c y y c x y 2 c x x − 2 ( c x x + c y y ) c x x c x y 2 c y y − c x y 2 ] {\displaystyle D_{Aniso}={\begin{bmatrix}{\frac {-c_{xy}}{2}}&c_{yy}&{\frac {c_{xy}}{2}}\\c_{xx}&-2(c_{xx}+c_{yy})&c_{xx}\\{\frac {c_{xy}}{2}}&c_{yy}&{\frac {-c_{xy}}{2}}\end{bmatrix}}} If, for example, c1=c2=1, the cxy component will vanish, resulting in the simple five-point stencil, rendering no controlled anisotropy. If c2>c1 and θ {\displaystyle \theta } =0, the anisotropic effects will be more pronounced in the vertical axis. If c2>c1 and θ {\displaystyle \theta } =45 degrees, the anisotropic effects will be more pronounced in the upper-right / lower-left diagonal. == References ==
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Wikipedia:Nira Dyn#0
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Nira (Richter) Dyn (Hebrew: נירה דין; born 1942) is an Israeli mathematician who studied geometric modeling, subdivision surfaces, approximation theory, and image compression. She is a professor emeritus of applied mathematics at Tel Aviv University, and has been called a "pioneer and leading researcher in the subdivision community". == Education and career == Dyn earned a bachelor's degree from the Technion – Israel Institute of Technology in 1965. She went on to graduate study at the Weizmann Institute of Science, where she earned a master's degree in 1967 and completed her doctorate in 1970. Her dissertation, Optimal and Minimum Norm Approximations to Linear Functionals in Hilbert Spaces, and their application to Numerical Integration, was supervised by Philip Rabinowitz. After postdoctoral research in the Institute of Fundamental Studies at the University of Rochester, she joined the Tel Aviv faculty in 1972, and retired in 2010. == Recognition == Dyn was an invited speaker at the 2006 International Congress of Mathematicians, in the section on numerical analysis and scientific computing. == Books == Dyn is the author of: Stochastic Models in Biology (as Nira Richter-Dyn, with Narendra S. Goel, Academic Press, 1974) Approximation of Set-valued Functions: Adaptation of Classical Approximation Operators (with Elza Farkhi and Alona Mokhov, Imperial College Press, 2014) == References == == External links == Home page
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Wikipedia:Nissan Deliatitz#0
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Nissan ben Avraham Deliatitz (Hebrew: ניסן בן אברהם דעליאטיץ) was a 19th-century Russian rabbi and mathematician. He wrote Keneh Ḥokhmah, a manual of algebra in five parts, published in Vilna and Grodno in 1829. The work received approbations from Rabbi David, the av beit din of Novhardok, and Rabbi Avraham Abele ben Avraham Shlomo Poswoler, an eminent scholar who headed the Vilna beit din. == References == This article incorporates text from a publication now in the public domain: Ginzberg, Louis; Seligsohn, M. (1901–1906). "Deliatitz, Nissan". In Singer, Isidore; et al. (eds.). The Jewish Encyclopedia. New York: Funk & Wagnalls.
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Wikipedia:Niven's theorem#0
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In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of θ in the interval 0° ≤ θ ≤ 90° for which the sine of θ degrees is also a rational number are: sin 0 ∘ = 0 , sin 30 ∘ = 1 2 , sin 90 ∘ = 1. {\displaystyle {\begin{aligned}\sin 0^{\circ }&=0,\\[10pt]\sin 30^{\circ }&={\frac {1}{2}},\\[10pt]\sin 90^{\circ }&=1.\end{aligned}}} In radians, one would require that 0° ≤ x ≤ π/2, that x/π be rational, and that sin(x) be rational. The conclusion is then that the only such values are sin(0) = 0, sin(π/6) = 1/2, and sin(π/2) = 1. The theorem appears as Corollary 3.12 in Niven's book on irrational numbers. The theorem extends to the other trigonometric functions as well. For rational values of θ, the only rational values of the sine or cosine are 0, ±1/2, and ±1; the only rational values of the secant or cosecant are ±1 and ±2; and the only rational values of the tangent or cotangent are 0 and ±1. == History == Niven's proof of his theorem appears in his book Irrational Numbers. Earlier, the theorem had been proven by D. H. Lehmer and J. M. H. Olmstead. In his 1933 paper, Lehmer proved the theorem for the cosine by proving a more general result. Namely, Lehmer showed that for relatively prime integers k and n with n > 2, the number 2 cos(2πk/n) is an algebraic number of degree φ(n)/2, where φ denotes Euler's totient function. Because rational numbers have degree 1, we must have n ≤ 2 or φ(n) = 2 and therefore the only possibilities are n = 1,2,3,4,6. Next, he proved a corresponding result for the sine using the trigonometric identity sin(θ) = cos(θ − π/2). In 1956, Niven extended Lehmer's result to the other trigonometric functions. Other mathematicians have given new proofs in subsequent years. == See also == Pythagorean triples form right triangles where the trigonometric functions will always take rational values, though the acute angles are not rational. Trigonometric functions Trigonometric number == References == == Further reading == Olmsted, J. M. H. (1945). "Rational values of trigonometric functions". The American Mathematical Monthly. 52 (9): 507–508. doi:10.2307/2304540. JSTOR 2304540. Jahnel, Jörg (2010). "When is the (co)sine of a rational angle equal to a rational number?". arXiv:1006.2938 [math.HO]. == External links == Weisstein, Eric W. "Niven's Theorem". MathWorld.
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Wikipedia:Nizar Touzi#0
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Nizar Touzi (born 1968 in Tunisia) is a Tunisian-French mathematician. He is a professor of applied mathematics at École polytechnique. His research focuses on analysis, statistics and algebra. He is being known for publications on optimization and stochastic control. == Education == Touzi completed his PhD in Applied Mathematics at the Paris Dauphine University under Éric Michel Renault in January 1994. He began his post-doctoral studies at the University of Chicago, doing such from October 1993 to May 1994. After this, he had an HDR at his alma mater, Paris Dauphine University, in January 1999. == Career == Touzi began his academic career as an assistant professor at this same institution in September 1994. He worked there for five years before becoming a professor of applied mathematics at Pantheone-Sorbonne University in Paris in September 1999. Touzi’s most cited work, Applications of Malliavin Calculus to Monte Carlo Methods in Finance, was published right before this career change in August 1999. In 2001, Touzi transitioned to the Center for Research in Economics and Statistics to continue teaching applied mathematics. Along with teaching, he also co-led the Finance and Insurance Laboratory at CREST. Between 2001 and 2005, Touzi was an invited professor at multiple institutions, including the University of British Columbia, Princeton University, and the Center for Interuniversity Research and Analysis of Organizations. In September 2005, Touzi accepted a new position as the Chair in Mathematical Finance at the Tanaka Business School of Imperial College London. He worked at the Tanaka Business School for almost a year before holding his most recent and current position as a professor of applied mathematics at École polytechnique. He was also the head of the Department of Applied Mathematics at École polytechnique from September 2014 to August 2017. == Research == Touzi's most cited paper, Applications of Malliavin Calculus to Monte Carlo methods in finance, co-authored by Eric Fournié, Jean-Michel Lasry, Jérôme Lebuchoux and Pierre-Louis Lions, describes an original probabilistic method to compute option contract Greeks: delta, gamma, theta, and vega. The method is derived from the formula for integration-by-parts and uses principles from Malliavin calculus. Their approach, when computed on standard European option contracts and compared to results yielded from the Monte Carlo method, happens to be more efficient. This paper had a significant impact in the world of mathematical finance, as previous option contract pricing models were based around the Black-Scholes model and Monte Carlo simulations. == Awards == Best Young Researcher in Finance Award 2007 of the Europlace Institute of Finance. The University of Toronto Dean’s Distinguished Visitor Chair, Fields Institute, April-June 2010. Invited Session Speaker at the International Congress of Mathematics, August 2010, Hyderabad (India). ERC Advanced Grant 2012. French Academy of Science Bachelier Prize 2012. Oxford University Visiting Man Chair, One month within the period January July 2014. Minerva Lectures at Columbia University, October 2013, New York. Monetary Authority of Singapore (MAS) Visiting Professor, National University of Singapore, January-February 2018, Singapore. Van Eenam, Butcher, and Butcher Financial/Actuarial Faculty Lecture, the University of Michigan, Department of Mathematics, April 2018, Ann Arbor. == References ==
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Wikipedia:Noah Dana-Picard#0
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Noah Dana-Picard (born May 6, 1954) is an Israeli mathematician, professor and Talmudic scholar who has been the president of the Jerusalem College of Technology (JCT) since 2009. == Life == Born in France, Dana-Picard holds two PhDs; the first from Nice University, France (1981) and the second from Bar Ilan University in Israel (1990). He is also a Talmudic scholar and speaks four languages. Dana-Picard has taught at JCT for two decades and published more than 70 scientific articles in algebra, infinitesimal calculus and geometry, as well as many articles in Jewish law, philosophy and the Bible. He is also an expert in technology-based mathematics education. Dana-Picard believes in contributing to the community and has built his professional career on the synthesis of Jewish studies and higher education. On this topic, he stated that: My children, my students, and indeed all the students who study at JCT, have been taught that it is a religious value to obtain a full-scale academic and professional education. They know that Jewish history and Jewish values dictate that they must serve the Jewish people and the state of Israel. Their job is to build and strengthen this country, as scientists, engineers, accountants, businessmen, nurses, educators and, at the same time, Torah scholars. JCT teaches this wholesome approach, seeking to imbue its students with the morals and ethics of Jewish tradition alongside expertise in their chosen technological profession. In addition to his position at JCT, Dana-Picard also sits on the editorial board of two prestigious journals in Europe, The International Journal of Mathematics Education in Science and Technology and The International Journal for Technology in Mathematics Education, and has served as an advisor for several academic institutions in Israel, including the Weizmann Institute of Science and the Haifa University. Besides his academic activities, Dana-Picard is the spiritual leader of his congregation in Jerusalem, and is active there both in education and assistance to new immigrants. == Published works == Th. Dana-Picard, G. Mann and N. Zehavi (2011): From conic intersections to toric intersections: the case of the isoptic curves Th. Dana-Picard (2007): Motivating constraints of a pedagogy embedded Computer Algebra System, International Journal of Science and Mathematics Education 5 (2), 217-235; G. Mann, Th. Dana-Picard and N. Zehavi (2007): Technological Discourse on CAS-based Operative Knowledge, International Journal of Technology in Mathematics Education 14 (3), 113-120. Th. Dana-Picard (2004): Three-fold activities for discovering conceptual connections within the cognitive neighborhood of a mathematical topic, Proceedings of TIME-2004 (ACDCA symposium) in Montreal (Canada), Journal B¨ohm (ed.), CD, bk teachware Schriftenreihe 41, Linz, Austria. Th. Dana-Picard (2005): Technology assisted discovery of conceptual connections within the cognitive neighborhood of a mathematical topic, Proceedings of CERME 4, M. Bosch (ed), San Feliu de Guixols (Spain). Th. Dana-Picard and I. Kidron (2006): A pedagogy-embedded Computer Algebra System as an instigator to learn more Mathematics, Proceedings of the ICMI Study 17 Conference, Hanoi, Vietnam, 2006. Th. Dana-Picard, I. Kidron and D. Zeitoun (2007): Strange 3D plots, Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education, D. Pitta-Pantazi and G. Philipou (edt.) Larnaca, 1379-1388. Th. Dana-Picard, G. Mann and N. Zehavi (2012): Instrumented techniques and extension of curriculum in Analytic Geometry, Proceedings of ICME-12, Seoul, South Korea, 3796-3804. Full list of publications: http://ndp.jct.ac.il/publications/home.html == References ==
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Wikipedia:Noether Lecture#0
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The Noether Lecture is a distinguished lecture series that honors women "who have made fundamental and sustained contributions to the mathematical sciences". The Association for Women in Mathematics (AWM) established the annual lectures in 1980 as the Emmy Noether Lectures, in honor of one of the leading mathematicians of her time. In 2013 it was renamed the AWM-AMS Noether Lecture and since 2015 is sponsored jointly with the American Mathematical Society (AMS). The recipient delivers the lecture at the yearly American Joint Mathematics Meetings held in January. The ICM Emmy Noether Lecture is an additional lecture series, sponsored by the International Mathematical Union. Beginning in 1994 this lecture was delivered at the International Congress of Mathematicians, held every four years. In 2010 the lecture series was made permanent. The 2021 Noether Lecture was supposed to have been given by Andrea Bertozzi of UCLA, but it was cancelled. The cancellation was made during the George Floyd protests: "This decision comes as many of this nation rise up in protest over racial discrimination and brutality by police". Although she intended to speak on other topics, Bertozzi is known for research on the mathematics of policing, and in a letter to the AMS, Sol Garfunkel concluded that "the reason for her exclusion was one of her areas of research". In an official blog of the AMS, a group calling themselves The Just Mathematics Collective called for a boycott of mathematical collaborations with police, dismissing Garfunkel's letter as "intended to further dismiss the boycott" and celebrating the cancellation of Bertozzi's lecture. == Noether Lecturer == == ICM Emmy Noether Lecturers == == See also == Falconer Lecture Kovalevsky Lecture List of mathematics awards List of things named after Emmy Noether == References == == External links == Official website
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Wikipedia:Noether identities#0
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In mathematics, Noether identities characterize the degeneracy of a Lagrangian system. Given a Lagrangian system and its Lagrangian L, Noether identities can be defined as a differential operator whose kernel contains a range of the Euler–Lagrange operator of L. Any Euler–Lagrange operator obeys Noether identities which therefore are separated into the trivial and non-trivial ones. A Lagrangian L is called degenerate if the Euler–Lagrange operator of L satisfies non-trivial Noether identities. In this case Euler–Lagrange equations are not independent. Noether identities need not be independent, but satisfy first-stage Noether identities, which are subject to the second-stage Noether identities and so on. Higher-stage Noether identities also are separated into trivial and non-trivial cases. A degenerate Lagrangian is called reducible if there exist non-trivial higher-stage Noether identities. Yang–Mills gauge theory and gauge gravitation theory exemplify irreducible Lagrangian field theories. Different variants of second Noether’s theorem state the one-to-one correspondence between the non-trivial reducible Noether identities and the non-trivial reducible gauge symmetries. Formulated in a very general setting, second Noether’s theorem associates to the Koszul–Tate complex of reducible Noether identities, parameterized by antifields, the BRST complex of reducible gauge symmetries parameterized by ghosts. This is the case of covariant classical field theory and Lagrangian BRST theory. == See also == Noether's second theorem Emmy Noether Lagrangian system Variational bicomplex Gauge symmetry (mathematics) == References == Gomis, J., Paris, J., Samuel, S., Antibracket, antifields and gauge theory quantization, Phys. Rep. 259 (1995) 1. Fulp, R., Lada, T., Stasheff, J. Noether variational theorem II and the BV formalism, arXiv:math/0204079 Bashkirov, D., Giachetta, G., Mangiarotti, L., Sardanashvily, G., The KT-BRST complex of a degenerate Lagrangian system, Lett. Math. Phys. 83 (2008) 237; arXiv:math-ph/0702097. Sardanashvily, G., Noether theorems in a general setting, arXiv:1411.2910.
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Wikipedia:Noetherian#0
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In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite length. Noetherian objects are named after Emmy Noether, who was the first to study the ascending and descending chain conditions for rings. Specifically: Noetherian group, a group that satisfies the ascending chain condition on subgroups. Noetherian ring, a ring that satisfies the ascending chain condition on ideals. Noetherian module, a module that satisfies the ascending chain condition on submodules. More generally, an object in a category is said to be Noetherian if there is no infinitely increasing filtration of it by subobjects. A category is Noetherian if every object in it is Noetherian. Noetherian relation, a binary relation that satisfies the ascending chain condition on its elements. Noetherian topological space, a topological space that satisfies the descending chain condition on closed sets. Noetherian induction, also called well-founded induction, a proof method for binary relations that satisfy the descending chain condition. Noetherian rewriting system, an abstract rewriting system that has no infinite chains. Noetherian scheme, a scheme in algebraic geometry that admits a finite covering by open spectra of Noetherian rings. == See also == Artinian ring, a ring that satisfies the descending chain condition on ideals.
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Wikipedia:Nola Anderson Haynes#0
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Nola Anderson Haynes (1897–1996) was an American mathematician and one of the few women to earn her PhD in math in the United States before World War II. == Biography == Nola Lee Anderson was born January 9, 1897, on a farm in 1897 in Linn County, Missouri, as one of four children. Her early education took place in a one-room schoolhouse; she graduated from the high school in Bucklin, Missouri in 1915 and found a job teaching school. In the fall of 1919, Anderson enrolled at the University of Missouri at age 22, and graduated three years later with a BS in education. In 1922, she started teaching high school and after two years and she took a job teaching mathematics at the Central College for Women in Lexington, Missouri. She returned to the University of Missouri in Columbia, for her master's degree with a major in mathematics and a minor in astronomy. She received her doctorate in 1929, in mathematics and astronomy, under Louis Ingold (1872–1935) with the dissertation An Extension of Maschke's Symbolism. One member of her advisory board was the head of the Astronomy department, Eli Stuart Haynes, who would later become her husband. In 1930, Anderson joined the faculty at H. Sophie Newcomb Memorial College of Tulane University in New Orleans, as associate professor and acting chair of the department. She left Newcomb in 1938 to get married, and moved back to Columbia, Missouri. In 1943, she returned to the University of Missouri, where she taught in the mathematics department where few women taught at the time. She was hired as an "acting" associate professor because nepotism laws at the time restricted her employment at the University. She later said, "It was not until my husband retired that I got the appointment of associate professor." She remained an associate professor at the University of Missouri from 1946 until in 1967 when she retired at 70, as emeritus associate professor . == Personal life == On July 9, 1938, Anderson and Haynes were married. Nola Haynes became a widow in 1956. In memory of her late husband, she established the Eli Stuart Haynes and Nola Anderson Haynes Scholarship Fund. In 1995, the Department of Mathematics and the College of Arts & Science presented her with the first Silver Chalk Award. Nola Anderson Haynes died in Brookfield, Missouri, on December 21, 1996, less than three weeks before her 100th birthday. She was buried in Memorial Park Cemetery, Columbia, Missouri. == Selected publications == She was the author of three articles on her research in geometry: "An Extension of Maschke's Symbolism" in the American Journal of Mathematics, 1929 "The Trigonometry of Hyperspace" in The American Mathematical Monthly, 1929 "Normals to a Space V n in Hyperspace" in the Bulletin of the American Mathematical Society, 1936 == Memberships == According to Judy Green, Haynes was active in several societies. President, League of Women Voters, Columbia, 1939–1941 National Society Colonial Dames American (Role of Honor award 1980) Sigma Xi Pi Lambda Theta Pi Mu Epsilon == References ==
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Wikipedia:Non-negative matrix factorization#0
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Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting matrices easier to inspect. Also, in applications such as processing of audio spectrograms or muscular activity, non-negativity is inherent to the data being considered. Since the problem is not exactly solvable in general, it is commonly approximated numerically. NMF finds applications in such fields as astronomy, computer vision, document clustering, missing data imputation, chemometrics, audio signal processing, recommender systems, and bioinformatics. == History == In chemometrics non-negative matrix factorization has a long history under the name "self modeling curve resolution". In this framework the vectors in the right matrix are continuous curves rather than discrete vectors. Also early work on non-negative matrix factorizations was performed by a Finnish group of researchers in the 1990s under the name positive matrix factorization. It became more widely known as non-negative matrix factorization after Lee and Seung investigated the properties of the algorithm and published some simple and useful algorithms for two types of factorizations. == Background == Let matrix V be the product of the matrices W and H, V = W H . {\displaystyle \mathbf {V} =\mathbf {W} \mathbf {H} \,.} Matrix multiplication can be implemented as computing the column vectors of V as linear combinations of the column vectors in W using coefficients supplied by columns of H. That is, each column of V can be computed as follows: v i = W h i , {\displaystyle \mathbf {v} _{i}=\mathbf {W} \mathbf {h} _{i}\,,} where vi is the i-th column vector of the product matrix V and hi is the i-th column vector of the matrix H. When multiplying matrices, the dimensions of the factor matrices may be significantly lower than those of the product matrix and it is this property that forms the basis of NMF. NMF generates factors with significantly reduced dimensions compared to the original matrix. For example, if V is an m × n matrix, W is an m × p matrix, and H is a p × n matrix then p can be significantly less than both m and n. Here is an example based on a text-mining application: Let the input matrix (the matrix to be factored) be V with 10000 rows and 500 columns where words are in rows and documents are in columns. That is, we have 500 documents indexed by 10000 words. It follows that a column vector v in V represents a document. Assume we ask the algorithm to find 10 features in order to generate a features matrix W with 10000 rows and 10 columns and a coefficients matrix H with 10 rows and 500 columns. The product of W and H is a matrix with 10000 rows and 500 columns, the same shape as the input matrix V and, if the factorization worked, it is a reasonable approximation to the input matrix V. From the treatment of matrix multiplication above it follows that each column in the product matrix WH is a linear combination of the 10 column vectors in the features matrix W with coefficients supplied by the coefficients matrix H. This last point is the basis of NMF because we can consider each original document in our example as being built from a small set of hidden features. NMF generates these features. It is useful to think of each feature (column vector) in the features matrix W as a document archetype comprising a set of words where each word's cell value defines the word's rank in the feature: The higher a word's cell value the higher the word's rank in the feature. A column in the coefficients matrix H represents an original document with a cell value defining the document's rank for a feature. We can now reconstruct a document (column vector) from our input matrix by a linear combination of our features (column vectors in W) where each feature is weighted by the feature's cell value from the document's column in H. == Clustering property == NMF has an inherent clustering property, i.e., it automatically clusters the columns of input data V = ( v 1 , … , v n ) {\displaystyle \mathbf {V} =(v_{1},\dots ,v_{n})} . More specifically, the approximation of V {\displaystyle \mathbf {V} } by V ≃ W H {\displaystyle \mathbf {V} \simeq \mathbf {W} \mathbf {H} } is achieved by finding W {\displaystyle W} and H {\displaystyle H} that minimize the error function (using the Frobenius norm) ‖ V − W H ‖ F , {\displaystyle \left\|V-WH\right\|_{F},} subject to W ≥ 0 , H ≥ 0. {\displaystyle W\geq 0,H\geq 0.} , If we furthermore impose an orthogonality constraint on H {\displaystyle \mathbf {H} } , i.e. H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} , then the above minimization is mathematically equivalent to the minimization of K-means clustering. Furthermore, the computed H {\displaystyle H} gives the cluster membership, i.e., if H k j > H i j {\displaystyle \mathbf {H} _{kj}>\mathbf {H} _{ij}} for all i ≠ k, this suggests that the input data v j {\displaystyle v_{j}} belongs to k {\displaystyle k} -th cluster. The computed W {\displaystyle W} gives the cluster centroids, i.e., the k {\displaystyle k} -th column gives the cluster centroid of k {\displaystyle k} -th cluster. This centroid's representation can be significantly enhanced by convex NMF. When the orthogonality constraint H H T = I {\displaystyle \mathbf {H} \mathbf {H} ^{T}=I} is not explicitly imposed, the orthogonality holds to a large extent, and the clustering property holds too. When the error function to be used is Kullback–Leibler divergence, NMF is identical to the probabilistic latent semantic analysis (PLSA), a popular document clustering method. == Types == === Approximate non-negative matrix factorization === Usually the number of columns of W and the number of rows of H in NMF are selected so the product WH will become an approximation to V. The full decomposition of V then amounts to the two non-negative matrices W and H as well as a residual U, such that: V = WH + U. The elements of the residual matrix can either be negative or positive. When W and H are smaller than V they become easier to store and manipulate. Another reason for factorizing V into smaller matrices W and H, is that if one's goal is to approximately represent the elements of V by significantly less data, then one has to infer some latent structure in the data. === Convex non-negative matrix factorization === In standard NMF, matrix factor W ∈ R+m × k， i.e., W can be anything in that space. Convex NMF restricts the columns of W to convex combinations of the input data vectors ( v 1 , … , v n ) {\displaystyle (v_{1},\dots ,v_{n})} . This greatly improves the quality of data representation of W. Furthermore, the resulting matrix factor H becomes more sparse and orthogonal. === Nonnegative rank factorization === In case the nonnegative rank of V is equal to its actual rank, V = WH is called a nonnegative rank factorization (NRF). The problem of finding the NRF of V, if it exists, is known to be NP-hard. === Different cost functions and regularizations === There are different types of non-negative matrix factorizations. The different types arise from using different cost functions for measuring the divergence between V and WH and possibly by regularization of the W and/or H matrices. Two simple divergence functions studied by Lee and Seung are the squared error (or Frobenius norm) and an extension of the Kullback–Leibler divergence to positive matrices (the original Kullback–Leibler divergence is defined on probability distributions). Each divergence leads to a different NMF algorithm, usually minimizing the divergence using iterative update rules. The factorization problem in the squared error version of NMF may be stated as: Given a matrix V {\displaystyle \mathbf {V} } find nonnegative matrices W and H that minimize the function F ( W , H ) = ‖ V − W H ‖ F 2 {\displaystyle F(\mathbf {W} ,\mathbf {H} )=\left\|\mathbf {V} -\mathbf {WH} \right\|_{F}^{2}} Another type of NMF for images is based on the total variation norm. When L1 regularization (akin to Lasso) is added to NMF with the mean squared error cost function, the resulting problem may be called non-negative sparse coding due to the similarity to the sparse coding problem, although it may also still be referred to as NMF. === Online NMF === Many standard NMF algorithms analyze all the data together; i.e., the whole matrix is available from the start. This may be unsatisfactory in applications where there are too many data to fit into memory or where the data are provided in streaming fashion. One such use is for collaborative filtering in recommendation systems, where there may be many users and many items to recommend, and it would be inefficient to recalculate everything when one user or one item is added to the system. The cost function for optimization in these cases may or may not be the same as for standard NMF, but the algorithms need to be rather different. === Convolutional NMF === If the columns of V represent data sampled over spatial or temporal dimensions, e.g. time signals, images, or video, features that are equivariant w.r.t. shifts along these dimensions can be learned by Convolutional NMF. In this case, W is sparse with columns having local non-zero weight windows that are shared across shifts along the spatio-temporal dimensions of V, representing convolution kernels. By spatio-temporal pooling of H and repeatedly using the resulting representation as input to convolutional NMF, deep feature hierarchies can be learned. == Algorithms == There are several ways in which the W and H may be found: Lee and Seung's multiplicative update rule has been a popular method due to the simplicity of implementation. This algorithm is: initialize: W and H non negative. Then update the values in W and H by computing the following, with n {\displaystyle n} as an index of the iteration. H [ i , j ] n + 1 ← H [ i , j ] n ( ( W n ) T V ) [ i , j ] ( ( W n ) T W n H n ) [ i , j ] {\displaystyle \mathbf {H} _{[i,j]}^{n+1}\leftarrow \mathbf {H} _{[i,j]}^{n}{\frac {((\mathbf {W} ^{n})^{T}\mathbf {V} )_{[i,j]}}{((\mathbf {W} ^{n})^{T}\mathbf {W} ^{n}\mathbf {H} ^{n})_{[i,j]}}}} and W [ i , j ] n + 1 ← W [ i , j ] n ( V ( H n + 1 ) T ) [ i , j ] ( W n H n + 1 ( H n + 1 ) T ) [ i , j ] {\displaystyle \mathbf {W} _{[i,j]}^{n+1}\leftarrow \mathbf {W} _{[i,j]}^{n}{\frac {(\mathbf {V} (\mathbf {H} ^{n+1})^{T})_{[i,j]}}{(\mathbf {W} ^{n}\mathbf {H} ^{n+1}(\mathbf {H} ^{n+1})^{T})_{[i,j]}}}} Until W and H are stable. Note that the updates are done on an element by element basis not matrix multiplication. We note that the multiplicative factors for W and H, i.e. the W T V W T W H {\textstyle {\frac {\mathbf {W} ^{\mathsf {T}}\mathbf {V} }{\mathbf {W} ^{\mathsf {T}}\mathbf {W} \mathbf {H} }}} and V H T W H H T {\textstyle {\textstyle {\frac {\mathbf {V} \mathbf {H} ^{\mathsf {T}}}{\mathbf {W} \mathbf {H} \mathbf {H} ^{\mathsf {T}}}}}} terms, are matrices of ones when V = W H {\displaystyle \mathbf {V} =\mathbf {W} \mathbf {H} } . More recently other algorithms have been developed. Some approaches are based on alternating non-negative least squares: in each step of such an algorithm, first H is fixed and W found by a non-negative least squares solver, then W is fixed and H is found analogously. The procedures used to solve for W and H may be the same or different, as some NMF variants regularize one of W and H. Specific approaches include the projected gradient descent methods, the active set method, the optimal gradient method, and the block principal pivoting method among several others. Current algorithms are sub-optimal in that they only guarantee finding a local minimum, rather than a global minimum of the cost function. A provably optimal algorithm is unlikely in the near future as the problem has been shown to generalize the k-means clustering problem which is known to be NP-complete. However, as in many other data mining applications, a local minimum may still prove to be useful. In addition to the optimization step, initialization has a significant effect on NMF. The initial values chosen for W and H may affect not only the rate of convergence, but also the overall error at convergence. Some options for initialization include complete randomization, SVD, k-means clustering, and more advanced strategies based on these and other paradigms. === Sequential NMF === The sequential construction of NMF components (W and H) was firstly used to relate NMF with Principal Component Analysis (PCA) in astronomy. The contribution from the PCA components are ranked by the magnitude of their corresponding eigenvalues; for NMF, its components can be ranked empirically when they are constructed one by one (sequentially), i.e., learn the ( n + 1 ) {\displaystyle (n+1)} -th component with the first n {\displaystyle n} components constructed. The contribution of the sequential NMF components can be compared with the Karhunen–Loève theorem, an application of PCA, using the plot of eigenvalues. A typical choice of the number of components with PCA is based on the "elbow" point, then the existence of the flat plateau is indicating that PCA is not capturing the data efficiently, and at last there exists a sudden drop reflecting the capture of random noise and falls into the regime of overfitting. For sequential NMF, the plot of eigenvalues is approximated by the plot of the fractional residual variance curves, where the curves decreases continuously, and converge to a higher level than PCA, which is the indication of less over-fitting of sequential NMF. === Exact NMF === Exact solutions for the variants of NMF can be expected (in polynomial time) when additional constraints hold for matrix V. A polynomial time algorithm for solving nonnegative rank factorization if V contains a monomial sub matrix of rank equal to its rank was given by Campbell and Poole in 1981. Kalofolias and Gallopoulos (2012) solved the symmetric counterpart of this problem, where V is symmetric and contains a diagonal principal sub matrix of rank r. Their algorithm runs in O(rm2) time in the dense case. Arora, Ge, Halpern, Mimno, Moitra, Sontag, Wu, & Zhu (2013) give a polynomial time algorithm for exact NMF that works for the case where one of the factors W satisfies a separability condition. == Relation to other techniques == In Learning the parts of objects by non-negative matrix factorization Lee and Seung proposed NMF mainly for parts-based decomposition of images. It compares NMF to vector quantization and principal component analysis, and shows that although the three techniques may be written as factorizations, they implement different constraints and therefore produce different results. It was later shown that some types of NMF are an instance of a more general probabilistic model called "multinomial PCA". When NMF is obtained by minimizing the Kullback–Leibler divergence, it is in fact equivalent to another instance of multinomial PCA, probabilistic latent semantic analysis, trained by maximum likelihood estimation. That method is commonly used for analyzing and clustering textual data and is also related to the latent class model. NMF with the least-squares objective is equivalent to a relaxed form of K-means clustering: the matrix factor W contains cluster centroids and H contains cluster membership indicators. This provides a theoretical foundation for using NMF for data clustering. However, k-means does not enforce non-negativity on its centroids, so the closest analogy is in fact with "semi-NMF". NMF can be seen as a two-layer directed graphical model with one layer of observed random variables and one layer of hidden random variables. NMF extends beyond matrices to tensors of arbitrary order. This extension may be viewed as a non-negative counterpart to, e.g., the PARAFAC model. Other extensions of NMF include joint factorization of several data matrices and tensors where some factors are shared. Such models are useful for sensor fusion and relational learning. NMF is an instance of nonnegative quadratic programming, just like the support vector machine (SVM). However, SVM and NMF are related at a more intimate level than that of NQP, which allows direct application of the solution algorithms developed for either of the two methods to problems in both domains. == Uniqueness == The factorization is not unique: A matrix and its inverse can be used to transform the two factorization matrices by, e.g., W H = W B B − 1 H {\displaystyle \mathbf {WH} =\mathbf {WBB} ^{-1}\mathbf {H} } If the two new matrices W ~ = W B {\displaystyle \mathbf {{\tilde {W}}=WB} } and H ~ = B − 1 H {\displaystyle \mathbf {\tilde {H}} =\mathbf {B} ^{-1}\mathbf {H} } are non-negative they form another parametrization of the factorization. The non-negativity of W ~ {\displaystyle \mathbf {\tilde {W}} } and H ~ {\displaystyle \mathbf {\tilde {H}} } applies at least if B is a non-negative monomial matrix. In this simple case it will just correspond to a scaling and a permutation. More control over the non-uniqueness of NMF is obtained with sparsity constraints. == Applications == === Astronomy === In astronomy, NMF is a promising method for dimension reduction in the sense that astrophysical signals are non-negative. NMF has been applied to the spectroscopic observations and the direct imaging observations as a method to study the common properties of astronomical objects and post-process the astronomical observations. The advances in the spectroscopic observations by Blanton & Roweis (2007) takes into account of the uncertainties of astronomical observations, which is later improved by Zhu (2016) where missing data are also considered and parallel computing is enabled. Their method is then adopted by Ren et al. (2018) to the direct imaging field as one of the methods of detecting exoplanets, especially for the direct imaging of circumstellar disks. Ren et al. (2018) are able to prove the stability of NMF components when they are constructed sequentially (i.e., one by one), which enables the linearity of the NMF modeling process; the linearity property is used to separate the stellar light and the light scattered from the exoplanets and circumstellar disks. In direct imaging, to reveal the faint exoplanets and circumstellar disks from bright the surrounding stellar lights, which has a typical contrast from 10⁵ to 10¹⁰, various statistical methods have been adopted, however the light from the exoplanets or circumstellar disks are usually over-fitted, where forward modeling have to be adopted to recover the true flux. Forward modeling is currently optimized for point sources, however not for extended sources, especially for irregularly shaped structures such as circumstellar disks. In this situation, NMF has been an excellent method, being less over-fitting in the sense of the non-negativity and sparsity of the NMF modeling coefficients, therefore forward modeling can be performed with a few scaling factors, rather than a computationally intensive data re-reduction on generated models. === Data imputation === To impute missing data in statistics, NMF can take missing data while minimizing its cost function, rather than treating these missing data as zeros. This makes it a mathematically proven method for data imputation in statistics. By first proving that the missing data are ignored in the cost function, then proving that the impact from missing data can be as small as a second order effect, Ren et al. (2020) studied and applied such an approach for the field of astronomy. Their work focuses on two-dimensional matrices, specifically, it includes mathematical derivation, simulated data imputation, and application to on-sky data. The data imputation procedure with NMF can be composed of two steps. First, when the NMF components are known, Ren et al. (2020) proved that impact from missing data during data imputation ("target modeling" in their study) is a second order effect. Second, when the NMF components are unknown, the authors proved that the impact from missing data during component construction is a first-to-second order effect. Depending on the way that the NMF components are obtained, the former step above can be either independent or dependent from the latter. In addition, the imputation quality can be increased when the more NMF components are used, see Figure 4 of Ren et al. (2020) for their illustration. === Text mining === NMF can be used for text mining applications. In this process, a document-term matrix is constructed with the weights of various terms (typically weighted word frequency information) from a set of documents. This matrix is factored into a term-feature and a feature-document matrix. The features are derived from the contents of the documents, and the feature-document matrix describes data clusters of related documents. One specific application used hierarchical NMF on a small subset of scientific abstracts from PubMed. Another research group clustered parts of the Enron email dataset with 65,033 messages and 91,133 terms into 50 clusters. NMF has also been applied to citations data, with one example clustering English Wikipedia articles and scientific journals based on the outbound scientific citations in English Wikipedia. Arora, Ge, Halpern, Mimno, Moitra, Sontag, Wu, & Zhu (2013) have given polynomial-time algorithms to learn topic models using NMF. The algorithm assumes that the topic matrix satisfies a separability condition that is often found to hold in these settings. Hassani, Iranmanesh and Mansouri (2019) proposed a feature agglomeration method for term-document matrices which operates using NMF. The algorithm reduces the term-document matrix into a smaller matrix more suitable for text clustering. === Spectral data analysis === NMF is also used to analyze spectral data; one such use is in the classification of space objects and debris. === Scalable Internet distance prediction === NMF is applied in scalable Internet distance (round-trip time) prediction. For a network with N {\displaystyle N} hosts, with the help of NMF, the distances of all the N 2 {\displaystyle N^{2}} end-to-end links can be predicted after conducting only O ( N ) {\displaystyle O(N)} measurements. This kind of method was firstly introduced in Internet Distance Estimation Service (IDES). Afterwards, as a fully decentralized approach, Phoenix network coordinate system is proposed. It achieves better overall prediction accuracy by introducing the concept of weight. === Non-stationary speech denoising === Speech denoising has been a long lasting problem in audio signal processing. There are many algorithms for denoising if the noise is stationary. For example, the Wiener filter is suitable for additive Gaussian noise. However, if the noise is non-stationary, the classical denoising algorithms usually have poor performance because the statistical information of the non-stationary noise is difficult to estimate. Schmidt et al. use NMF to do speech denoising under non-stationary noise, which is completely different from classical statistical approaches. The key idea is that clean speech signal can be sparsely represented by a speech dictionary, but non-stationary noise cannot. Similarly, non-stationary noise can also be sparsely represented by a noise dictionary, but speech cannot. The algorithm for NMF denoising goes as follows. Two dictionaries, one for speech and one for noise, need to be trained offline. Once a noisy speech is given, we first calculate the magnitude of the Short-Time-Fourier-Transform. Second, separate it into two parts via NMF, one can be sparsely represented by the speech dictionary, and the other part can be sparsely represented by the noise dictionary. Third, the part that is represented by the speech dictionary will be the estimated clean speech. === Population genetics === Sparse NMF is used in Population genetics for estimating individual admixture coefficients, detecting genetic clusters of individuals in a population sample or evaluating genetic admixture in sampled genomes. In human genetic clustering, NMF algorithms provide estimates similar to those of the computer program STRUCTURE, but the algorithms are more efficient computationally and allow analysis of large population genomic data sets. === Bioinformatics === NMF has been successfully applied in bioinformatics for clustering gene expression and DNA methylation data and finding the genes most representative of the clusters. In the analysis of cancer mutations it has been used to identify common patterns of mutations that occur in many cancers and that probably have distinct causes. NMF techniques can identify sources of variation such as cell types, disease subtypes, population stratification, tissue composition, and tumor clonality. A particular variant of NMF, namely Non-Negative Matrix Tri-Factorization (NMTF), has been use for drug repurposing tasks in order to predict novel protein targets and therapeutic indications for approved drugs and to infer pair of synergic anticancer drugs. === Nuclear imaging === NMF, also referred in this field as factor analysis, has been used since the 1980s to analyze sequences of images in SPECT and PET dynamic medical imaging. Non-uniqueness of NMF was addressed using sparsity constraints. == Current research == Current research (since 2010) in nonnegative matrix factorization includes, but is not limited to, Algorithmic: searching for global minima of the factors and factor initialization. Scalability: how to factorize million-by-billion matrices, which are commonplace in Web-scale data mining, e.g., see Distributed Nonnegative Matrix Factorization (DNMF), Scalable Nonnegative Matrix Factorization (ScalableNMF), Distributed Stochastic Singular Value Decomposition. Online: how to update the factorization when new data comes in without recomputing from scratch, e.g., see online CNSC Collective (joint) factorization: factorizing multiple interrelated matrices for multiple-view learning, e.g. multi-view clustering, see CoNMF and MultiNMF Cohen and Rothblum 1993 problem: whether a rational matrix always has an NMF of minimal inner dimension whose factors are also rational. Recently, this problem has been answered negatively. == See also == Multilinear algebra Multilinear subspace learning Tensor Tensor decomposition Tensor software == Sources and external links == === Notes === === Others ===
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Wikipedia:Nonlinear eigenproblem#0
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In mathematics, a nonlinear eigenproblem, sometimes nonlinear eigenvalue problem, is a generalization of the (ordinary) eigenvalue problem to equations that depend nonlinearly on the eigenvalue. Specifically, it refers to equations of the form M ( λ ) x = 0 , {\displaystyle M(\lambda )x=0,} where x ≠ 0 {\displaystyle x\neq 0} is a vector, and M {\displaystyle M} is a matrix-valued function of the number λ {\displaystyle \lambda } . The number λ {\displaystyle \lambda } is known as the (nonlinear) eigenvalue, the vector x {\displaystyle x} as the (nonlinear) eigenvector, and ( λ , x ) {\displaystyle (\lambda ,x)} as the eigenpair. The matrix M ( λ ) {\displaystyle M(\lambda )} is singular at an eigenvalue λ {\displaystyle \lambda } . == Definition == In the discipline of numerical linear algebra the following definition is typically used. Let Ω ⊆ C {\displaystyle \Omega \subseteq \mathbb {C} } , and let M : Ω → C n × n {\displaystyle M:\Omega \rightarrow \mathbb {C} ^{n\times n}} be a function that maps scalars to matrices. A scalar λ ∈ C {\displaystyle \lambda \in \mathbb {C} } is called an eigenvalue, and a nonzero vector x ∈ C n {\displaystyle x\in \mathbb {C} ^{n}} is called a right eigevector if M ( λ ) x = 0 {\displaystyle M(\lambda )x=0} . Moreover, a nonzero vector y ∈ C n {\displaystyle y\in \mathbb {C} ^{n}} is called a left eigevector if y H M ( λ ) = 0 H {\displaystyle y^{H}M(\lambda )=0^{H}} , where the superscript H {\displaystyle ^{H}} denotes the Hermitian transpose. The definition of the eigenvalue is equivalent to det ( M ( λ ) ) = 0 {\displaystyle \det(M(\lambda ))=0} , where det ( ) {\displaystyle \det()} denotes the determinant. The function M {\displaystyle M} is usually required to be a holomorphic function of λ {\displaystyle \lambda } (in some domain Ω {\displaystyle \Omega } ). In general, M ( λ ) {\displaystyle M(\lambda )} could be a linear map, but most commonly it is a finite-dimensional, usually square, matrix. Definition: The problem is said to be regular if there exists a z ∈ Ω {\displaystyle z\in \Omega } such that det ( M ( z ) ) ≠ 0 {\displaystyle \det(M(z))\neq 0} . Otherwise it is said to be singular. Definition: An eigenvalue λ {\displaystyle \lambda } is said to have algebraic multiplicity k {\displaystyle k} if k {\displaystyle k} is the smallest integer such that the k {\displaystyle k} th derivative of det ( M ( z ) ) {\displaystyle \det(M(z))} with respect to z {\displaystyle z} , in λ {\displaystyle \lambda } is nonzero. In formulas that d k det ( M ( z ) ) d z k | z = λ ≠ 0 {\displaystyle \left.{\frac {d^{k}\det(M(z))}{dz^{k}}}\right|_{z=\lambda }\neq 0} but d ℓ det ( M ( z ) ) d z ℓ | z = λ = 0 {\displaystyle \left.{\frac {d^{\ell }\det(M(z))}{dz^{\ell }}}\right|_{z=\lambda }=0} for ℓ = 0 , 1 , 2 , … , k − 1 {\displaystyle \ell =0,1,2,\dots ,k-1} . Definition: The geometric multiplicity of an eigenvalue λ {\displaystyle \lambda } is the dimension of the nullspace of M ( λ ) {\displaystyle M(\lambda )} . == Special cases == The following examples are special cases of the nonlinear eigenproblem. The (ordinary) eigenvalue problem: M ( λ ) = A − λ I . {\displaystyle M(\lambda )=A-\lambda I.} The generalized eigenvalue problem: M ( λ ) = A − λ B . {\displaystyle M(\lambda )=A-\lambda B.} The quadratic eigenvalue problem: M ( λ ) = A 0 + λ A 1 + λ 2 A 2 . {\displaystyle M(\lambda )=A_{0}+\lambda A_{1}+\lambda ^{2}A_{2}.} The polynomial eigenvalue problem: M ( λ ) = ∑ i = 0 m λ i A i . {\displaystyle M(\lambda )=\sum _{i=0}^{m}\lambda ^{i}A_{i}.} The rational eigenvalue problem: M ( λ ) = ∑ i = 0 m 1 A i λ i + ∑ i = 1 m 2 B i r i ( λ ) , {\displaystyle M(\lambda )=\sum _{i=0}^{m_{1}}A_{i}\lambda ^{i}+\sum _{i=1}^{m_{2}}B_{i}r_{i}(\lambda ),} where r i ( λ ) {\displaystyle r_{i}(\lambda )} are rational functions. The delay eigenvalue problem: M ( λ ) = − I λ + A 0 + ∑ i = 1 m A i e − τ i λ , {\displaystyle M(\lambda )=-I\lambda +A_{0}+\sum _{i=1}^{m}A_{i}e^{-\tau _{i}\lambda },} where τ 1 , τ 2 , … , τ m {\displaystyle \tau _{1},\tau _{2},\dots ,\tau _{m}} are given scalars, known as delays. == Jordan chains == Definition: Let ( λ 0 , x 0 ) {\displaystyle (\lambda _{0},x_{0})} be an eigenpair. A tuple of vectors ( x 0 , x 1 , … , x r − 1 ) ∈ C n × C n × ⋯ × C n {\displaystyle (x_{0},x_{1},\dots ,x_{r-1})\in \mathbb {C} ^{n}\times \mathbb {C} ^{n}\times \dots \times \mathbb {C} ^{n}} is called a Jordan chain if ∑ k = 0 ℓ M ( k ) ( λ 0 ) x ℓ − k = 0 , {\displaystyle \sum _{k=0}^{\ell }M^{(k)}(\lambda _{0})x_{\ell -k}=0,} for ℓ = 0 , 1 , … , r − 1 {\displaystyle \ell =0,1,\dots ,r-1} , where M ( k ) ( λ 0 ) {\displaystyle M^{(k)}(\lambda _{0})} denotes the k {\displaystyle k} th derivative of M {\displaystyle M} with respect to λ {\displaystyle \lambda } and evaluated in λ = λ 0 {\displaystyle \lambda =\lambda _{0}} . The vectors x 0 , x 1 , … , x r − 1 {\displaystyle x_{0},x_{1},\dots ,x_{r-1}} are called generalized eigenvectors, r {\displaystyle r} is called the length of the Jordan chain, and the maximal length a Jordan chain starting with x 0 {\displaystyle x_{0}} is called the rank of x 0 {\displaystyle x_{0}} . Theorem: A tuple of vectors ( x 0 , x 1 , … , x r − 1 ) ∈ C n × C n × ⋯ × C n {\displaystyle (x_{0},x_{1},\dots ,x_{r-1})\in \mathbb {C} ^{n}\times \mathbb {C} ^{n}\times \dots \times \mathbb {C} ^{n}} is a Jordan chain if and only if the function M ( λ ) χ ℓ ( λ ) {\displaystyle M(\lambda )\chi _{\ell }(\lambda )} has a root in λ = λ 0 {\displaystyle \lambda =\lambda _{0}} and the root is of multiplicity at least ℓ {\displaystyle \ell } for ℓ = 0 , 1 , … , r − 1 {\displaystyle \ell =0,1,\dots ,r-1} , where the vector valued function χ ℓ ( λ ) {\displaystyle \chi _{\ell }(\lambda )} is defined as χ ℓ ( λ ) = ∑ k = 0 ℓ x k ( λ − λ 0 ) k . {\displaystyle \chi _{\ell }(\lambda )=\sum _{k=0}^{\ell }x_{k}(\lambda -\lambda _{0})^{k}.} == Mathematical software == The eigenvalue solver package SLEPc contains C-implementations of many numerical methods for nonlinear eigenvalue problems. The NLEVP collection of nonlinear eigenvalue problems is a MATLAB package containing many nonlinear eigenvalue problems with various properties. The FEAST eigenvalue solver is a software package for standard eigenvalue problems as well as nonlinear eigenvalue problems, designed from density-matrix representation in quantum mechanics combined with contour integration techniques. The MATLAB toolbox NLEIGS contains an implementation of fully rational Krylov with a dynamically constructed rational interpolant. The MATLAB toolbox CORK contains an implementation of the compact rational Krylov algorithm that exploits the Kronecker structure of the linearization pencils. The MATLAB toolbox AAA-EIGS contains an implementation of CORK with rational approximation by set-valued AAA. The MATLAB toolbox RKToolbox (Rational Krylov Toolbox) contains implementations of the rational Krylov method for nonlinear eigenvalue problems as well as features for rational approximation. The Julia package NEP-PACK contains many implementations of various numerical methods for nonlinear eigenvalue problems, as well as many benchmark problems. The review paper of Güttel & Tisseur contains MATLAB code snippets implementing basic Newton-type methods and contour integration methods for nonlinear eigenproblems. == Eigenvector nonlinearity == Eigenvector nonlinearities is a related, but different, form of nonlinearity that is sometimes studied. In this case the function M {\displaystyle M} maps vectors to matrices, or sometimes hermitian matrices to hermitian matrices. == References == == Further reading == Françoise Tisseur and Karl Meerbergen, "The quadratic eigenvalue problem," SIAM Review 43 (2), 235–286 (2001) (link). Gene H. Golub and Henk A. van der Vorst, "Eigenvalue computation in the 20th century," Journal of Computational and Applied Mathematics 123, 35–65 (2000). Philippe Guillaume, "Nonlinear eigenproblems," SIAM Journal on Matrix Analysis and Applications 20 (3), 575–595 (1999) (link). Cedric Effenberger, "Robust solution methods fornonlinear eigenvalue problems", PhD thesis EPFL (2013) (link) Roel Van Beeumen, "Rational Krylov methods fornonlinear eigenvalue problems", PhD thesis KU Leuven (2015) (link)
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Wikipedia:Nonlocal operator#0
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In mathematics, a nonlocal operator is a mapping which maps functions on a topological space to functions, in such a way that the value of the output function at a given point cannot be determined solely from the values of the input function in any neighbourhood of any point. An example of a nonlocal operator is the Fourier transform. == Formal definition == Let X {\displaystyle X} be a topological space, Y {\displaystyle Y} a set, F ( X ) {\displaystyle F(X)} a function space containing functions with domain X {\displaystyle X} , and G ( Y ) {\displaystyle G(Y)} a function space containing functions with domain Y {\displaystyle Y} . Two functions u {\displaystyle u} and v {\displaystyle v} in F ( X ) {\displaystyle F(X)} are called equivalent at x ∈ X {\displaystyle x\in X} if there exists a neighbourhood N {\displaystyle N} of x {\displaystyle x} such that u ( x ′ ) = v ( x ′ ) {\displaystyle u(x')=v(x')} for all x ′ ∈ N {\displaystyle x'\in N} . An operator A : F ( X ) → G ( Y ) {\displaystyle A:F(X)\to G(Y)} is said to be local if for every y ∈ Y {\displaystyle y\in Y} there exists an x ∈ X {\displaystyle x\in X} such that A u ( y ) = A v ( y ) {\displaystyle Au(y)=Av(y)} for all functions u {\displaystyle u} and v {\displaystyle v} in F ( X ) {\displaystyle F(X)} which are equivalent at x {\displaystyle x} . A nonlocal operator is an operator which is not local. For a local operator it is possible (in principle) to compute the value A u ( y ) {\displaystyle Au(y)} using only knowledge of the values of u {\displaystyle u} in an arbitrarily small neighbourhood of a point x {\displaystyle x} . For a nonlocal operator this is not possible. == Examples == Differential operators are examples of local operators. A large class of (linear) nonlocal operators is given by the integral transforms, such as the Fourier transform and the Laplace transform. For an integral transform of the form ( A u ) ( y ) = ∫ X u ( x ) K ( x , y ) d x , {\displaystyle (Au)(y)=\int \limits _{X}u(x)\,K(x,y)\,dx,} where K {\displaystyle K} is some kernel function, it is necessary to know the values of u {\displaystyle u} almost everywhere on the support of K ( ⋅ , y ) {\displaystyle K(\cdot ,y)} in order to compute the value of A u {\displaystyle Au} at y {\displaystyle y} . An example of a singular integral operator is the fractional Laplacian ( − Δ ) s f ( x ) = c d , s ∫ R d f ( x ) − f ( y ) | x − y | d + 2 s d y . {\displaystyle (-\Delta )^{s}f(x)=c_{d,s}\int \limits _{\mathbb {R} ^{d}}{\frac {f(x)-f(y)}{|x-y|^{d+2s}}}\,dy.} The prefactor c d , s := 4 s Γ ( d / 2 + s ) π d / 2 | Γ ( − s ) | {\displaystyle c_{d,s}:={\frac {4^{s}\Gamma (d/2+s)}{\pi ^{d/2}|\Gamma (-s)|}}} involves the Gamma function and serves as a normalizing factor. The fractional Laplacian plays a role in, for example, the study of nonlocal minimal surfaces. == Applications == Some examples of applications of nonlocal operators are: Time series analysis using Fourier transformations Analysis of dynamical systems using Laplace transformations Image denoising using non-local means Modelling Gaussian blur or motion blur in images using convolution with a blurring kernel or point spread function == See also == Fractional calculus Linear map Nonlocal Lagrangian Action at a distance == References == == External links == Nonlocal equations wiki
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Wikipedia:Nonnegative rank (linear algebra)#0
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In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the "nondegenerateness" of the system of linear equations and linear transformation encoded by A. There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics. The rank is commonly denoted by rank(A) or rk(A); sometimes the parentheses are not written, as in rank A. == Main definitions == In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these. The column rank of A is the dimension of the column space of A, while the row rank of A is the dimension of the row space of A. A fundamental result in linear algebra is that the column rank and the row rank are always equal. (Three proofs of this result are given in § Proofs that column rank = row rank, below.) This number (i.e., the number of linearly independent rows or columns) is simply called the rank of A. A matrix is said to have full rank if its rank equals the largest possible for a matrix of the same dimensions, which is the lesser of the number of rows and columns. A matrix is said to be rank-deficient if it does not have full rank. The rank deficiency of a matrix is the difference between the lesser of the number of rows and columns, and the rank. The rank of a linear map or operator Φ {\displaystyle \Phi } is defined as the dimension of its image: rank ( Φ ) := dim ( img ( Φ ) ) {\displaystyle \operatorname {rank} (\Phi ):=\dim(\operatorname {img} (\Phi ))} where dim {\displaystyle \dim } is the dimension of a vector space, and img {\displaystyle \operatorname {img} } is the image of a map. == Examples == The matrix [ 1 0 1 0 1 1 0 1 1 ] {\displaystyle {\begin{bmatrix}1&0&1\\0&1&1\\0&1&1\end{bmatrix}}} has rank 2: the first two columns are linearly independent, so the rank is at least 2, but since the third is a linear combination of the first two (the first column plus the second), the three columns are linearly dependent so the rank must be less than 3. The matrix A = [ 1 1 0 2 − 1 − 1 0 − 2 ] {\displaystyle A={\begin{bmatrix}1&1&0&2\\-1&-1&0&-2\end{bmatrix}}} has rank 1: there are nonzero columns, so the rank is positive, but any pair of columns is linearly dependent. Similarly, the transpose A T = [ 1 − 1 1 − 1 0 0 2 − 2 ] {\displaystyle A^{\mathrm {T} }={\begin{bmatrix}1&-1\\1&-1\\0&0\\2&-2\end{bmatrix}}} of A has rank 1. Indeed, since the column vectors of A are the row vectors of the transpose of A, the statement that the column rank of a matrix equals its row rank is equivalent to the statement that the rank of a matrix is equal to the rank of its transpose, i.e., rank(A) = rank(AT). == Computing the rank of a matrix == === Rank from row echelon forms === A common approach to finding the rank of a matrix is to reduce it to a simpler form, generally row echelon form, by elementary row operations. Row operations do not change the row space (hence do not change the row rank), and, being invertible, map the column space to an isomorphic space (hence do not change the column rank). Once in row echelon form, the rank is clearly the same for both row rank and column rank, and equals the number of pivots (or basic columns) and also the number of non-zero rows. For example, the matrix A given by A = [ 1 2 1 − 2 − 3 1 3 5 0 ] {\displaystyle A={\begin{bmatrix}1&2&1\\-2&-3&1\\3&5&0\end{bmatrix}}} can be put in reduced row-echelon form by using the following elementary row operations: [ 1 2 1 − 2 − 3 1 3 5 0 ] → 2 R 1 + R 2 → R 2 [ 1 2 1 0 1 3 3 5 0 ] → − 3 R 1 + R 3 → R 3 [ 1 2 1 0 1 3 0 − 1 − 3 ] → R 2 + R 3 → R 3 [ 1 2 1 0 1 3 0 0 0 ] → − 2 R 2 + R 1 → R 1 [ 1 0 − 5 0 1 3 0 0 0 ] . {\displaystyle {\begin{aligned}{\begin{bmatrix}1&2&1\\-2&-3&1\\3&5&0\end{bmatrix}}&\xrightarrow {2R_{1}+R_{2}\to R_{2}} {\begin{bmatrix}1&2&1\\0&1&3\\3&5&0\end{bmatrix}}\xrightarrow {-3R_{1}+R_{3}\to R_{3}} {\begin{bmatrix}1&2&1\\0&1&3\\0&-1&-3\end{bmatrix}}\\&\xrightarrow {R_{2}+R_{3}\to R_{3}} \,\,{\begin{bmatrix}1&2&1\\0&1&3\\0&0&0\end{bmatrix}}\xrightarrow {-2R_{2}+R_{1}\to R_{1}} {\begin{bmatrix}1&0&-5\\0&1&3\\0&0&0\end{bmatrix}}~.\end{aligned}}} The final matrix (in reduced row echelon form) has two non-zero rows and thus the rank of matrix A is 2. === Computation === When applied to floating point computations on computers, basic Gaussian elimination (LU decomposition) can be unreliable, and a rank-revealing decomposition should be used instead. An effective alternative is the singular value decomposition (SVD), but there are other less computationally expensive choices, such as QR decomposition with pivoting (so-called rank-revealing QR factorization), which are still more numerically robust than Gaussian elimination. Numerical determination of rank requires a criterion for deciding when a value, such as a singular value from the SVD, should be treated as zero, a practical choice which depends on both the matrix and the application. == Proofs that column rank = row rank == === Proof using row reduction === The fact that the column and row ranks of any matrix are equal forms is fundamental in linear algebra. Many proofs have been given. One of the most elementary ones has been sketched in § Rank from row echelon forms. Here is a variant of this proof: It is straightforward to show that neither the row rank nor the column rank are changed by an elementary row operation. As Gaussian elimination proceeds by elementary row operations, the reduced row echelon form of a matrix has the same row rank and the same column rank as the original matrix. Further elementary column operations allow putting the matrix in the form of an identity matrix possibly bordered by rows and columns of zeros. Again, this changes neither the row rank nor the column rank. It is immediate that both the row and column ranks of this resulting matrix is the number of its nonzero entries. We present two other proofs of this result. The first uses only basic properties of linear combinations of vectors, and is valid over any field. The proof is based upon Wardlaw (2005). The second uses orthogonality and is valid for matrices over the real numbers; it is based upon Mackiw (1995). Both proofs can be found in the book by Banerjee and Roy (2014). === Proof using linear combinations === Let A be an m × n matrix. Let the column rank of A be r, and let c1, ..., cr be any basis for the column space of A. Place these as the columns of an m × r matrix C. Every column of A can be expressed as a linear combination of the r columns in C. This means that there is an r × n matrix R such that A = CR. R is the matrix whose ith column is formed from the coefficients giving the ith column of A as a linear combination of the r columns of C. In other words, R is the matrix which contains the multiples for the bases of the column space of A (which is C), which are then used to form A as a whole. Now, each row of A is given by a linear combination of the r rows of R. Therefore, the rows of R form a spanning set of the row space of A and, by the Steinitz exchange lemma, the row rank of A cannot exceed r. This proves that the row rank of A is less than or equal to the column rank of A. This result can be applied to any matrix, so apply the result to the transpose of A. Since the row rank of the transpose of A is the column rank of A and the column rank of the transpose of A is the row rank of A, this establishes the reverse inequality and we obtain the equality of the row rank and the column rank of A. (Also see Rank factorization.) === Proof using orthogonality === Let A be an m × n matrix with entries in the real numbers whose row rank is r. Therefore, the dimension of the row space of A is r. Let x1, x2, …, xr be a basis of the row space of A. We claim that the vectors Ax1, Ax2, …, Axr are linearly independent. To see why, consider a linear homogeneous relation involving these vectors with scalar coefficients c1, c2, …, cr: 0 = c 1 A x 1 + c 2 A x 2 + ⋯ + c r A x r = A ( c 1 x 1 + c 2 x 2 + ⋯ + c r x r ) = A v , {\displaystyle 0=c_{1}A\mathbf {x} _{1}+c_{2}A\mathbf {x} _{2}+\cdots +c_{r}A\mathbf {x} _{r}=A(c_{1}\mathbf {x} _{1}+c_{2}\mathbf {x} _{2}+\cdots +c_{r}\mathbf {x} _{r})=A\mathbf {v} ,} where v = c1x1 + c2x2 + ⋯ + crxr. We make two observations: (a) v is a linear combination of vectors in the row space of A, which implies that v belongs to the row space of A, and (b) since Av = 0, the vector v is orthogonal to every row vector of A and, hence, is orthogonal to every vector in the row space of A. The facts (a) and (b) together imply that v is orthogonal to itself, which proves that v = 0 or, by the definition of v, c 1 x 1 + c 2 x 2 + ⋯ + c r x r = 0. {\displaystyle c_{1}\mathbf {x} _{1}+c_{2}\mathbf {x} _{2}+\cdots +c_{r}\mathbf {x} _{r}=0.} But recall that the xi were chosen as a basis of the row space of A and so are linearly independent. This implies that c1 = c2 = ⋯ = cr = 0. It follows that Ax1, Ax2, …, Axr are linearly independent. Now, each Axi is obviously a vector in the column space of A. So, Ax1, Ax2, …, Axr is a set of r linearly independent vectors in the column space of A and, hence, the dimension of the column space of A (i.e., the column rank of A) must be at least as big as r. This proves that row rank of A is no larger than the column rank of A. Now apply this result to the transpose of A to get the reverse inequality and conclude as in the previous proof. == Alternative definitions == In all the definitions in this section, the matrix A is taken to be an m × n matrix over an arbitrary field F. === Dimension of image === Given the matrix A {\displaystyle A} , there is an associated linear mapping f : F n → F m {\displaystyle f:F^{n}\to F^{m}} defined by f ( x ) = A x . {\displaystyle f(x)=Ax.} The rank of A {\displaystyle A} is the dimension of the image of f {\displaystyle f} . This definition has the advantage that it can be applied to any linear map without need for a specific matrix. === Rank in terms of nullity === Given the same linear mapping f as above, the rank is n minus the dimension of the kernel of f. The rank–nullity theorem states that this definition is equivalent to the preceding one. === Column rank – dimension of column space === The rank of A is the maximal number of linearly independent columns c 1 , c 2 , … , c k {\displaystyle \mathbf {c} _{1},\mathbf {c} _{2},\dots ,\mathbf {c} _{k}} of A; this is the dimension of the column space of A (the column space being the subspace of Fm generated by the columns of A, which is in fact just the image of the linear map f associated to A). === Row rank – dimension of row space === The rank of A is the maximal number of linearly independent rows of A; this is the dimension of the row space of A. === Decomposition rank === The rank of A is the smallest positive integer k such that A can be factored as A = C R {\displaystyle A=CR} , where C is an m × k matrix and R is a k × n matrix. In fact, for all integers k, the following are equivalent: the column rank of A is less than or equal to k, there exist k columns c 1 , … , c k {\displaystyle \mathbf {c} _{1},\ldots ,\mathbf {c} _{k}} of size m such that every column of A is a linear combination of c 1 , … , c k {\displaystyle \mathbf {c} _{1},\ldots ,\mathbf {c} _{k}} , there exist an m × k {\displaystyle m\times k} matrix C and a k × n {\displaystyle k\times n} matrix R such that A = C R {\displaystyle A=CR} (when k is the rank, this is a rank factorization of A), there exist k rows r 1 , … , r k {\displaystyle \mathbf {r} _{1},\ldots ,\mathbf {r} _{k}} of size n such that every row of A is a linear combination of r 1 , … , r k {\displaystyle \mathbf {r} _{1},\ldots ,\mathbf {r} _{k}} , the row rank of A is less than or equal to k. Indeed, the following equivalences are obvious: ( 1 ) ⇔ ( 2 ) ⇔ ( 3 ) ⇔ ( 4 ) ⇔ ( 5 ) {\displaystyle (1)\Leftrightarrow (2)\Leftrightarrow (3)\Leftrightarrow (4)\Leftrightarrow (5)} . For example, to prove (3) from (2), take C to be the matrix whose columns are c 1 , … , c k {\displaystyle \mathbf {c} _{1},\ldots ,\mathbf {c} _{k}} from (2). To prove (2) from (3), take c 1 , … , c k {\displaystyle \mathbf {c} _{1},\ldots ,\mathbf {c} _{k}} to be the columns of C. It follows from the equivalence ( 1 ) ⇔ ( 5 ) {\displaystyle (1)\Leftrightarrow (5)} that the row rank is equal to the column rank. As in the case of the "dimension of image" characterization, this can be generalized to a definition of the rank of any linear map: the rank of a linear map f : V → W is the minimal dimension k of an intermediate space X such that f can be written as the composition of a map V → X and a map X → W. Unfortunately, this definition does not suggest an efficient manner to compute the rank (for which it is better to use one of the alternative definitions). See rank factorization for details. === Rank in terms of singular values === The rank of A equals the number of non-zero singular values, which is the same as the number of non-zero diagonal elements in Σ in the singular value decomposition A = U Σ V ∗ {\displaystyle A=U\Sigma V^{*}} . === Determinantal rank – size of largest non-vanishing minor === The rank of A is the largest order of any non-zero minor in A. (The order of a minor is the side-length of the square sub-matrix of which it is the determinant.) Like the decomposition rank characterization, this does not give an efficient way of computing the rank, but it is useful theoretically: a single non-zero minor witnesses a lower bound (namely its order) for the rank of the matrix, which can be useful (for example) to prove that certain operations do not lower the rank of a matrix. A non-vanishing p-minor (p × p submatrix with non-zero determinant) shows that the rows and columns of that submatrix are linearly independent, and thus those rows and columns of the full matrix are linearly independent (in the full matrix), so the row and column rank are at least as large as the determinantal rank; however, the converse is less straightforward. The equivalence of determinantal rank and column rank is a strengthening of the statement that if the span of n vectors has dimension p, then p of those vectors span the space (equivalently, that one can choose a spanning set that is a subset of the vectors): the equivalence implies that a subset of the rows and a subset of the columns simultaneously define an invertible submatrix (equivalently, if the span of n vectors has dimension p, then p of these vectors span the space and there is a set of p coordinates on which they are linearly independent). === Tensor rank – minimum number of simple tensors === The rank of A is the smallest number k such that A can be written as a sum of k rank 1 matrices, where a matrix is defined to have rank 1 if and only if it can be written as a nonzero product c ⋅ r {\displaystyle c\cdot r} of a column vector c and a row vector r. This notion of rank is called tensor rank; it can be generalized in the separable models interpretation of the singular value decomposition. == Properties == We assume that A is an m × n matrix, and we define the linear map f by f(x) = Ax as above. The rank of an m × n matrix is a nonnegative integer and cannot be greater than either m or n. That is, rank ( A ) ≤ min ( m , n ) . {\displaystyle \operatorname {rank} (A)\leq \min(m,n).} A matrix that has rank min(m, n) is said to have full rank; otherwise, the matrix is rank deficient. Only a zero matrix has rank zero. f is injective (or "one-to-one") if and only if A has rank n (in this case, we say that A has full column rank). f is surjective (or "onto") if and only if A has rank m (in this case, we say that A has full row rank). If A is a square matrix (i.e., m = n), then A is invertible if and only if A has rank n (that is, A has full rank). If B is any n × k matrix, then rank ( A B ) ≤ min ( rank ( A ) , rank ( B ) ) . {\displaystyle \operatorname {rank} (AB)\leq \min(\operatorname {rank} (A),\operatorname {rank} (B)).} If B is an n × k matrix of rank n, then rank ( A B ) = rank ( A ) . {\displaystyle \operatorname {rank} (AB)=\operatorname {rank} (A).} If C is an l × m matrix of rank m, then rank ( C A ) = rank ( A ) . {\displaystyle \operatorname {rank} (CA)=\operatorname {rank} (A).} The rank of A is equal to r if and only if there exists an invertible m × m matrix X and an invertible n × n matrix Y such that X A Y = [ I r 0 0 0 ] , {\displaystyle XAY={\begin{bmatrix}I_{r}&0\\0&0\end{bmatrix}},} where Ir denotes the r × r identity matrix and the three zero matrices have the sizes r × (n − r), (m − r) × r and (m − r) × (n − r). Sylvester’s rank inequality: if A is an m × n matrix and B is n × k, then rank ( A ) + rank ( B ) − n ≤ rank ( A B ) . {\displaystyle \operatorname {rank} (A)+\operatorname {rank} (B)-n\leq \operatorname {rank} (AB).} This is a special case of the next inequality. The inequality due to Frobenius: if AB, ABC and BC are defined, then rank ( A B ) + rank ( B C ) ≤ rank ( B ) + rank ( A B C ) . {\displaystyle \operatorname {rank} (AB)+\operatorname {rank} (BC)\leq \operatorname {rank} (B)+\operatorname {rank} (ABC).} Subadditivity: rank ( A + B ) ≤ rank ( A ) + rank ( B ) {\displaystyle \operatorname {rank} (A+B)\leq \operatorname {rank} (A)+\operatorname {rank} (B)} when A and B are of the same dimension. As a consequence, a rank-k matrix can be written as the sum of k rank-1 matrices, but not fewer. The rank of a matrix plus the nullity of the matrix equals the number of columns of the matrix. (This is the rank–nullity theorem.) If A is a matrix over the real numbers then the rank of A and the rank of its corresponding Gram matrix are equal. Thus, for real matrices rank ( A T A ) = rank ( A A T ) = rank ( A ) = rank ( A T ) . {\displaystyle \operatorname {rank} (A^{\mathrm {T} }A)=\operatorname {rank} (AA^{\mathrm {T} })=\operatorname {rank} (A)=\operatorname {rank} (A^{\mathrm {T} }).} This can be shown by proving equality of their null spaces. The null space of the Gram matrix is given by vectors x for which A T A x = 0. {\displaystyle A^{\mathrm {T} }A\mathbf {x} =0.} If this condition is fulfilled, we also have 0 = x T A T A x = | A x | 2 . {\displaystyle 0=\mathbf {x} ^{\mathrm {T} }A^{\mathrm {T} }A\mathbf {x} =\left|A\mathbf {x} \right|^{2}.} If A is a matrix over the complex numbers and A ¯ {\displaystyle {\overline {A}}} denotes the complex conjugate of A and A∗ the conjugate transpose of A (i.e., the adjoint of A), then rank ( A ) = rank ( A ¯ ) = rank ( A T ) = rank ( A ∗ ) = rank ( A ∗ A ) = rank ( A A ∗ ) . {\displaystyle \operatorname {rank} (A)=\operatorname {rank} ({\overline {A}})=\operatorname {rank} (A^{\mathrm {T} })=\operatorname {rank} (A^{*})=\operatorname {rank} (A^{*}A)=\operatorname {rank} (AA^{*}).} == Applications == One useful application of calculating the rank of a matrix is the computation of the number of solutions of a system of linear equations. According to the Rouché–Capelli theorem, the system is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. If on the other hand, the ranks of these two matrices are equal, then the system must have at least one solution. The solution is unique if and only if the rank equals the number of variables. Otherwise the general solution has k free parameters where k is the difference between the number of variables and the rank. In this case (and assuming the system of equations is in the real or complex numbers) the system of equations has infinitely many solutions. In control theory, the rank of a matrix can be used to determine whether a linear system is controllable, or observable. In the field of communication complexity, the rank of the communication matrix of a function gives bounds on the amount of communication needed for two parties to compute the function. == Generalization == There are different generalizations of the concept of rank to matrices over arbitrary rings, where column rank, row rank, dimension of column space, and dimension of row space of a matrix may be different from the others or may not exist. Thinking of matrices as tensors, the tensor rank generalizes to arbitrary tensors; for tensors of order greater than 2 (matrices are order 2 tensors), rank is very hard to compute, unlike for matrices. There is a notion of rank for smooth maps between smooth manifolds. It is equal to the linear rank of the derivative. == Matrices as tensors == Matrix rank should not be confused with tensor order, which is called tensor rank. Tensor order is the number of indices required to write a tensor, and thus matrices all have tensor order 2. More precisely, matrices are tensors of type (1,1), having one row index and one column index, also called covariant order 1 and contravariant order 1; see Tensor (intrinsic definition) for details. The tensor rank of a matrix can also mean the minimum number of simple tensors necessary to express the matrix as a linear combination, and that this definition does agree with matrix rank as here discussed. == See also == Matroid rank Nonnegative rank (linear algebra) Rank (differential topology) Multicollinearity Linear dependence == Notes == == References == == Sources == Axler, Sheldon (2015). Linear Algebra Done Right. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 978-3-319-11079-0. Halmos, Paul Richard (1974) [1958]. Finite-Dimensional Vector Spaces. Undergraduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-90093-4. Hefferon, Jim (2020). Linear Algebra (4th ed.). Orthogonal Publishing L3C. ISBN 978-1-944325-11-4. Katznelson, Yitzhak; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. American Mathematical Society. ISBN 978-0-8218-4419-9. Roman, Steven (2005). Advanced Linear Algebra. Undergraduate Texts in Mathematics (2nd ed.). Springer. ISBN 0-387-24766-1. Valenza, Robert J. (1993) [1951]. Linear Algebra: An Introduction to Abstract Mathematics. Undergraduate Texts in Mathematics (3rd ed.). Springer. ISBN 3-540-94099-5. == Further reading == Roger A. Horn and Charles R. Johnson (1985). Matrix Analysis. Cambridge University Press. ISBN 978-0-521-38632-6. Kaw, Autar K. Two Chapters from the book Introduction to Matrix Algebra: 1. Vectors [1] and System of Equations [2] Mike Brookes: Matrix Reference Manual. [3]
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Wikipedia:Norbert A'Campo#0
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Norbert A'Campo (born 27 April 1941) is a Swiss mathematician working on singularity theory. He earned a doctorate in 1972 from the University of Paris-Sud. In 1974 he was an invited speaker at the International Congress of Mathematicians, and in 1988 he was elected president of the Swiss Mathematical Society. In 2012 he became a fellow of the American Mathematical Society. == References == == External links ==
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Wikipedia:Norbert Ryska#0
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Norbert Ryska (born August 9, 1948, in Hau) is a German mathematician and museum director. Ryska worked from 1976 to 1992 as an employee of Nixdorf Computer AG in the R&D department. Until 1996 as managing director and project manager on behalf of the Nixdorf Foundations mainly responsible for the construction of the Heinz Nixdorf MuseumsForum (HNF). From 1996 to 2013, Ryska was the managing director for the museum and technology departments of the HNF. == Early life == Ryska was born in Hau on the Lower Rhine in 1948, the son of the technician and operating engineer Ferdinand Ryska. He attended high school in Kleve. After graduating from high school in 1967, he completed his military service, e.g. in the communications center of the 1st Corps of the German Armed Forces in Münster where he learned how to use encryption technology. In the winter semester of 1968/69, Ryska studied English and Romance languages in Münster and from the summer of 1969 he continued his studies in philology for three further terms at Bonn University. From the winter semester of 1970/71 he studied mathematics and computer science. The mathematician Friedrich Hirzebruch was one of his professors. In the summer of 1976, Ryska received a diploma in mathematics. During his studies he had part-time jobs for Inter Nationes, the German Bundestag and the Reuters news agency. On November 1, 1976, Ryska joined Nixdorf Computer AG (NCAG) in Paderborn. His most important work in the years that followed was a security system for ATMs. He was then responsible for the coordination of national and European funding projects in the NCAG Research and Development department and was responsible for public relations. After the merger of NCAG with the data and information technology division of Siemens AG to form Siemens Nixdorf Informationssysteme AG (SNI) on October 1, 1990, Ryska continued to oversee funding projects at the SNI. From April 1, 1992, he worked on the project of a computer museum. The Westphalia Foundation, founded by Heinz Nixdorf, appointed Norbert Ryska to the project team for a computer museum in 1992. On April 1, 1993, he became managing director of the non-profit Forum für Informationstechnik GmbH. With the exhibition designer Ludwig Thürmer and the architect Gerhard Diel, Ryska completed the transformation of the NCAG headquarters into the Heinz Nixdorf MuseumsForum. He led the team that selected exhibits and created texts, images and films. When it opened on October 24, 1996, he was one of the two managing directors alongside Theodor Rode and was responsible for the permanent exhibition, the media department and technical staff. From 1997 to 2017, his co-managing director Kurt Beiersdörfer was responsible for marketing, events and museum education as well as for several special exhibitions. Together with Wolfgang Back (WDR), Ryska realized three WDR computer nights at the HNF in 1998, 1999 and 2001. In 2001, Ryska created a section on the history of cryptology; in 2004 he created a cryptology archive. In 2002 he initiated the multimedia presentation “Wall of Fame”, which was finished in 2004. In the same year, Ryska was able to convince the Nixdorf Foundation Board for his concept of an extensive expansion and update of the permanent exhibition to acquire and implement the areas of "Interfaces-Kommunikation mit der Maschine" ("interface communication with the machine"), "Künstliche Intelligenz und Robotik" ("artificial intelligence and robotics"), "Mobile Kommunikation" ("mobile communication"), "Digitale Welt" ("digital world"), "Showroom – Technik von morgen" ("showroom – technology of tomorrow"). In 2007, Ryska and the HNF participated in the Bletchley Park Computer Museum's Cipher Event. In the same year he created an exhibition on software and computer science. Between 2008 and 2012 he organized “Zahlen, bitte!” (“numbers/pay, please!”) a special exhibition on mathematics, “Codes und Clowns” (“Codes and Clowns”) on Claude Shannon and “Genial & Geheim” ("Ingenious & Secret") on Alan Turing. August 30, 2013 was Ryska's last day at HNF. He then engaged in private historical research. == Founding initiator and promoter of the computer museum HNF == The founding process of the development of a "computer museum" initiated by the Nixdorf Foundations (1992), which was to include the collection and product history of Nixdorf Computer AG (1992–1996), was moderated to a large extent by Ryska. In cooperation and consultation with more than 100 national and international experts (from science, technology, education, design and politics, including Rolf Oberliesen), he tried to combine the development and design of a museum with a public forum, for which previously there have been no models worldwide in relation to existing modern technology or company museums. He took on the task of combining the design and implementation of this still vague founding idea for a museum, consisting of existing technical artefacts (special information and communication technology collections) with an open forum. He led a difficult conceptual coordination process between previous company interests and public ideas and social demands. With a group of interior architects, designers and multimedia programmers, he succeeded in connecting a technology museum and a forum in the design of 60 identified exhibition areas.: 32–43 Together with Ludwig Thürmer and Gerhard Diel, a coherent concept for these presentations in the building of the former Nixdorf headquarters in Paderborn was developed. Right from the start, the focus was on a specific conceptual openness of the museum as a background for further specialist exploration and current issues of the forum. An open, non-linear presentation concept was realized that presented both the before and after one another as well as the simultaneity in concrete historical and social contexts of technological development, also as a basis for further current questions in an interdisciplinary public forum. Here, in a “spatial dramaturgy”, a “experiential museum” was created with “contemporary surroundings” perspectively integrated via “time stations” and a “cultural-historical panorama wall”.: 144 This also appeared as a specific contribution to a socially and ecologically compatible design of technology within the framework of public technology policy dialogues about the future of industrial societies. The presentations, designed over two floors, met a recognized level of high quality standards in terms of content and functionality. Even before the opening of the MuseumsForum by then Federal Chancellor Helmut Kohl on October 24, 1996, the previous sponsoring company of the Westphalia Foundation renamed the computer museum to "HNF Heinz Nixdorf MuseumsForum GmbH" (1995), also as a consistent confirmation of Ryska's pursued interdisciplinary integration concept of technology presence and forum designed for dialogue. Ryska was appointed managing director of the HNF (1996–2013), where he was able to decisively advance the previous development concept and expand the subject areas. == Publications == Kryptographische Verfahren in der Datenverarbeitung. Springer-Verlag, Berlin 1980, co-written with Siegfried Herda ISBN 978-3-540-09900-0. Nixdorf Technologie-Broschüren für CeBIT. Self-publishing, Paderborn 1981–1989. PapierKunst – 365 × im HNF: Objekte von Dorothea Reese-Heim. W.V. Westfalia Druck GmbH, Paderborn 2002, ISBN 978-3-9805757-3-7. Heinz Nixdorf. Lebensbilder 1925 – 1986. Merkur Druck, Detmold 2007, co-written with Margret Schwarte-Amedick. Kurze Geschichte der Informatik, Friedrich L. Bauer (mit redaktioneller Unterstützung von Norbert Ryska). Wilhelm Fink Verlag, München 2007, 2. Auflage: Wilhelm Fink Verlag, München 2009, ISBN 978-3-7705-4379-3 Origins and Foundations of Computinig, Friedrich L. Bauer (with editorial assistance by Norbert Ryska). Springer Verlag, München 2010, ISBN 978-3-642-02992-9. Weltgeschichte der Kryptologie. Data CD of the HNF Heinz Nixdorf Museumsforum, Paderborn 2015. == Other publications == Norbert Ryska: Möglichkeiten der Datenverschlüsselung bei Geldausgabe-Automaten: Hast du was – weißt du was – kriegst du was. Newspaper article Computerwoche, 31. Oktober 1980. Norbert Ryska, Jochen Viehoff: Codes und Clowns. Claude Shannon – Jongleur der Wissenschaft. Zeitschrift Mitteilungen der Deutschen Mathematiker-Vereinigung (MDMV) 17, pp. 236–238, 2009. Norbert Ryska: Kalender für das Klever Land auf das Jahr 2011. Geschichte: "Berühmter Rechenmaschinenerfinder stammte aus Kleve" – Johann Helfrich von Müller. Boss-Verlag, Kleve 2010, ISBN 978-3-89413-011-4 William Aspray, Len Shustek, Norbert Ryska: Computer museum series – Great computing museums of the world, part one. Communications of the ACM (CACM) 53(1): pp. 43–46, 2010. Rainer Glaschick, Norbert Ryska: Alan Turing und Deutschland: Berührungspunkte. Informatik Spektrum 35(4): pp. 295–300, 2012. Norbert Ryska, Jochen Viehoff: The Heinz Nixdorf Museums Forum, Central Venue for the "History of Computing". HC, London 2013: pp. 47–52. Norbert Ryska: Kalender für das Klever Land auf das Jahr 2021. Story: "Knickebein-Leitstrahlen für die Luftschlacht um England kamen aus Kleve". Aschendorff Verlag, Münster 2020, ISBN 978-3-402-22436-6. == Reference section == == External links section == Nixdorf Stiftungen: Heinz Nixdorf Stiftung und Stiftung Westfalen Heinz Nixdorf MuseumsForum
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Wikipedia:Noreen Sher Akbar#0
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Noreen Sher Akbar is a Pakistani applied mathematician specializing in fluid dynamics. After obtaining her Ph.D. in 2012, from Quaid-i-Azam University, she joined the faculty at National University of Sciences & Technology, where she is head of the Department of Basic Sciences and Humanities. Akbar has won multiple honours for her research productivity including being listed as a Young Associate of the Pakistan Academy of Sciences. She was the 2012 winner of the Best Young Scientist award of the National Academy of Young Scientists, and the 2017 winner of the M. Raziuddin Siddiqi Prize and Gold Medal of the Pakistan Academy of Sciences. == References == == External links == Noreen Sher Akbar publications indexed by Google Scholar
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Wikipedia:Noriko H. Arai#0
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Noriko H. Arai (Japanese: 新井紀子, romanized: Arai Noriko, born 1962) is a Japanese researcher in mathematical logic and artificial intelligence, known for her work on a project to develop robots that can pass the entrance examinations for the University of Tokyo. She is a professor in the information and society research division of the National Institute of Informatics. == Education and career == Arai was born in Tokyo. She earned a law degree from Hitotsubashi University and then, in 1985, a mathematics degree magna cum laude from the University of Illinois at Urbana–Champaign. Her doctorate is from the Tokyo Institute of Technology. She joined the National Institute of Informatics in 2001. == Contributions == Arai's Todai Robot Project aims to build a robot that can pass the entrance examinations for the University of Tokyo (commonly known as Todai) by 2021. Arai became director of the project in 2011. At a 2017 TED Talk, she reported that her system could achieve a score better than 80% of the applicants to the university; however, this was still not a passing score. Arai sees the success of the project as evidence that human education should concentrate more on problem solving and creativity, and less on rote learning. Arai is also the founder of Researchmap, "the largest social network for researchers in Japan". She was one of 15 top artificial intelligence researchers invited by French president Emmanuel Macron to join him in March 2018 for the announcement of a major new French initiative for artificial intelligence research. == References == == Further reading == Living in the AI Era: Noriko Arai, Mathematician, NHK, June 7, 2018 == External links == ResearchMap profile
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Wikipedia:Norm residue isomorphism theorem#0
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In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime ℓ {\displaystyle \ell } and any natural number n {\displaystyle n} . John Milnor speculated that this theorem might be true for ℓ = 2 {\displaystyle \ell =2} and all n {\displaystyle n} , and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato and became known as the Bloch–Kato conjecture or the motivic Bloch–Kato conjecture to distinguish it from the Bloch–Kato conjecture on values of L-functions. The norm residue isomorphism theorem was proved by Vladimir Voevodsky using a number of highly innovative results of Markus Rost. == Statement == For any integer ℓ invertible in a field k {\displaystyle k} there is a map ∂ : k × → H 1 ( k , μ ℓ ) {\displaystyle \partial :k^{\times }\rightarrow H^{1}(k,\mu _{\ell })} where μ ℓ {\displaystyle \mu _{\ell }} denotes the Galois module of ℓ-th roots of unity in some separable closure of k. It induces an isomorphism k × / ( k × ) ℓ ≅ H 1 ( k , μ ℓ ) {\displaystyle k^{\times }/(k^{\times })^{\ell }\cong H^{1}(k,\mu _{\ell })} . The first hint that this is related to K-theory is that k × {\displaystyle k^{\times }} is the group K1(k). Taking the tensor products and applying the multiplicativity of étale cohomology yields an extension of the map ∂ {\displaystyle \partial } to maps: ∂ n : k × ⊗ ⋯ ⊗ k × → H e ´ t n ( k , μ ℓ ⊗ n ) . {\displaystyle \partial ^{n}:k^{\times }\otimes \cdots \otimes k^{\times }\rightarrow H_{\rm {{\acute {e}}t}}^{n}(k,\mu _{\ell }^{\otimes n}).} These maps have the property that, for every element a in k ∖ { 0 , 1 } {\displaystyle k\setminus \{0,1\}} , ∂ n ( … , a , … , 1 − a , … ) {\displaystyle \partial ^{n}(\ldots ,a,\ldots ,1-a,\ldots )} vanishes. This is the defining relation of Milnor K-theory. Specifically, Milnor K-theory is defined to be the graded parts of the ring: K ∗ M ( k ) = T ( k × ) / ( { a ⊗ ( 1 − a ) : a ∈ k ∖ { 0 , 1 } } ) , {\displaystyle K_{*}^{M}(k)=T(k^{\times })/(\{a\otimes (1-a)\colon a\in k\setminus \{0,1\}\}),} where T ( k × ) {\displaystyle T(k^{\times })} is the tensor algebra of the multiplicative group k × {\displaystyle k^{\times }} and the quotient is by the two-sided ideal generated by all elements of the form a ⊗ ( 1 − a ) {\displaystyle a\otimes (1-a)} . Therefore the map ∂ n {\displaystyle \partial ^{n}} factors through a map: ∂ n : K n M ( k ) → H e ´ t n ( k , μ ℓ ⊗ n ) . {\displaystyle \partial ^{n}\colon K_{n}^{M}(k)\to H_{\rm {{\acute {e}}t}}^{n}(k,\mu _{\ell }^{\otimes n}).} This map is called the Galois symbol or norm residue map. Because étale cohomology with mod-ℓ coefficients is an ℓ-torsion group, this map additionally factors through K n M ( k ) / ℓ {\displaystyle K_{n}^{M}(k)/\ell } . The norm residue isomorphism theorem (or Bloch–Kato conjecture) states that for a field k and an integer ℓ that is invertible in k, the norm residue map ∂ n : K n M ( k ) / ℓ → H e ´ t n ( k , μ ℓ ⊗ n ) {\displaystyle \partial ^{n}:K_{n}^{M}(k)/\ell \to H_{\rm {{\acute {e}}t}}^{n}(k,\mu _{\ell }^{\otimes n})} from Milnor K-theory mod-ℓ to étale cohomology is an isomorphism. The case ℓ = 2 is the Milnor conjecture, and the case n = 2 is the Merkurjev–Suslin theorem. == History == The étale cohomology of a field is identical to Galois cohomology, so the conjecture equates the ℓth cotorsion (the quotient by the subgroup of ℓ-divisible elements) of the Milnor K-group of a field k with the Galois cohomology of k with coefficients in the Galois module of ℓth roots of unity. The point of the conjecture is that there are properties that are easily seen for Milnor K-groups but not for Galois cohomology, and vice versa; the norm residue isomorphism theorem makes it possible to apply techniques applicable to the object on one side of the isomorphism to the object on the other side of the isomorphism. The case when n is 0 is trivial, and the case when n = 1 follows easily from Hilbert's Theorem 90. The case n = 2 and ℓ = 2 was proved by (Merkurjev 1981). An important advance was the case n = 2 and ℓ arbitrary. This case was proved by (Merkurjev & Suslin 1982) and is known as the Merkurjev–Suslin theorem. Later, Merkurjev and Suslin, and independently, Rost, proved the case n = 3 and ℓ = 2 (Merkurjev & Suslin 1991) (Rost 1986). The name "norm residue" originally referred to the Hilbert symbol ( a 1 , a 2 ) {\displaystyle (a_{1},a_{2})} , which takes values in the Brauer group of k (when the field contains all ℓ-th roots of unity). Its usage here is in analogy with standard local class field theory and is expected to be part of an (as yet undeveloped) "higher" class field theory. The norm residue isomorphism theorem implies the Quillen–Lichtenbaum conjecture. === History of the proof === Milnor's conjecture was proved by Vladimir Voevodsky. Later Voevodsky proved the general Bloch–Kato conjecture. The starting point for the proof is a series of conjectures due to Lichtenbaum (1983) and Beilinson (1987). They conjectured the existence of motivic complexes, complexes of sheaves whose cohomology was related to motivic cohomology. Among the conjectural properties of these complexes were three properties: one connecting their Zariski cohomology to Milnor's K-theory, one connecting their etale cohomology to cohomology with coefficients in the sheaves of roots of unity and one connecting their Zariski cohomology to their etale cohomology. These three properties implied, as a very special case, that the norm residue map should be an isomorphism. The essential characteristic of the proof is that it uses the induction on the "weight" (which equals the dimension of the cohomology group in the conjecture) where the inductive step requires knowing not only the statement of Bloch-Kato conjecture but the much more general statement that contains a large part of the Beilinson-Lichtenbaum conjectures. It often occurs in proofs by induction that the statement being proved has to be strengthened in order to prove the inductive step. In this case the strengthening that was needed required the development of a very large amount of new mathematics. The earliest proof of Milnor's conjecture is contained in a 1995 preprint of Voevodsky and is inspired by the idea that there should be algebraic analogs of Morava K-theory (these algebraic Morava K-theories were later constructed by Simone Borghesi). In a 1996 preprint, Voevodsky was able to remove Morava K-theory from the picture by introducing instead algebraic cobordisms and using some of their properties that were not proved at that time (these properties were proved later). The constructions of 1995 and 1996 preprints are now known to be correct but the first completed proof of Milnor's conjecture used a somewhat different scheme. It is also the scheme that the proof of the full Bloch–Kato conjecture follows. It was devised by Voevodsky a few months after the 1996 preprint appeared. Implementing this scheme required making substantial advances in the field of motivic homotopy theory as well as finding a way to build algebraic varieties with a specified list of properties. From the motivic homotopy theory the proof required the following: A construction of the motivic analog of the basic ingredient of the Spanier–Whitehead duality in the form of the motivic fundamental class as a morphism from the motivic sphere to the Thom space of the motivic normal bundle over a smooth projective algebraic variety. A construction of the motivic analog of the Steenrod algebra. A proof of the proposition stating that over a field of characteristic zero the motivic Steenrod algebra characterizes all bi-stable cohomology operations in the motivic cohomology. The first two constructions were developed by Voevodsky by 2003. Combined with the results that had been known since late 1980s, they were sufficient to reprove the Milnor conjecture. Also in 2003, Voevodsky published on the web a preprint that nearly contained a proof of the general theorem. It followed the original scheme but was missing the proofs of three statements. Two of these statements were related to the properties of the motivic Steenrod operations and required the third fact above, while the third one required then-unknown facts about "norm varieties". The properties that these varieties were required to have had been formulated by Voevodsky in 1997, and the varieties themselves had been constructed by Markus Rost in 1998–2003. The proof that they have the required properties was completed by Andrei Suslin and Seva Joukhovitski in 2006. The third fact above required the development of new techniques in motivic homotopy theory. The goal was to prove that a functor, which was not assumed to commute with limits or colimits, preserved weak equivalences between objects of a certain form. One of the main difficulties there was that the standard approach to the study of weak equivalences is based on Bousfield–Quillen factorization systems and model category structures, and these were inadequate. Other methods had to be developed, and this work was completed by Voevodsky only in 2008. In the course of developing these techniques, it became clear that the first statement used without proof in Voevodsky's 2003 preprint is false. The proof had to be modified slightly to accommodate the corrected form of that statement. While Voevodsky continued to work out the final details of the proofs of the main theorems about motivic Eilenberg–MacLane spaces, Charles Weibel invented an approach to correct the place in the proof that had to modified. Weibel also published in 2009 a paper that contained a summary of Voevodsky's constructions combined with the correction that he discovered. == Beilinson–Lichtenbaum conjecture == Let X be a smooth variety over a field containing 1 / ℓ {\displaystyle 1/\ell } . Beilinson and Lichtenbaum conjectured that the motivic cohomology group H p , q ( X , Z / ℓ ) {\displaystyle H^{p,q}(X,\mathbf {Z} /\ell )} is isomorphic to the étale cohomology group H e ´ t p ( X , μ ℓ ⊗ q ) {\displaystyle H_{\rm {{\acute {e}}t}}^{p}(X,\mu _{\ell }^{\otimes q})} when p≤q. This conjecture has now been proven, and is equivalent to the norm residue isomorphism theorem. == References == == Bibliography == Bloch, Spencer; Kato, Kazuya (1986). "p-adic etale cohomology". Publications Mathématiques de l'IHÉS. 63: 107–152. doi:10.1007/bf02831624. Borghesi, Simone (2000), Algebraic Morava K-theories and the higher degree formula, Preprint Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs. Vol. 124. Providence, RI: American Mathematical Society. ISBN 0-8218-4041-X. Zbl 1103.12002. Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001. Milnor, John (1970). "Algebraic K-theory and quadratic forms". Inventiones Mathematicae. 9 (4): 318–344. Bibcode:1970InMat...9..318M. doi:10.1007/bf01425486. Rost, Markus (1998). "Chain lemma for splitting fields of symbols". Srinivas, V. (2008). Algebraic K-theory. Modern Birkhäuser Classics (Paperback reprint of the 1996 2nd ed.). Boston, MA: Birkhäuser. ISBN 978-0-8176-4736-0. Zbl 1125.19300. Voevodsky, Vladimir (1995), Bloch-Kato conjecture for Z/2-coefficients and algebraic Morava K-theories, Preprint, CiteSeerX 10.1.1.154.922 Voevodsky, Vladimir (1996), The Milnor Conjecture, Preprint Voevodsky, Vladimir (2001), On 2-torsion in motivic cohomology, Preprint, arXiv:math/0107110, Bibcode:2001math......7110V Voevodsky, Vladimir (2003a), "Reduced power operations in motivic cohomology", Institut des Hautes Études Scientifiques. Publications Mathématiques, 98 (1): 1–57, arXiv:math/0107109, doi:10.1007/s10240-003-0009-z, ISSN 0073-8301, MR 2031198 Voevodsky, Vladimir (2003b), "Motivic cohomology with Z/2-coefficients", Institut des Hautes Études Scientifiques. Publications Mathématiques, 98 (1): 59–104, doi:10.1007/s10240-003-0010-6, ISSN 0073-8301, MR 2031199 Voevodsky, Vladimir (2008). "On motivic cohomology with Z/L coefficients". arXiv:0805.4430 [math.AG]. Weibel, Charles (2009). "The norm residue isomorphism theorem". Journal of Topology. 2 (2): 346–372. doi:10.1112/jtopol/jtp013. MR 2529300. Voevodsky, Vladimir (2011). "On motivic cohomology with Z/L-coefficients". Annals of Mathematics. 174 (1): 401–438. arXiv:0805.4430. doi:10.4007/annals.2011.174.1.11.
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Wikipedia:Norma Presmeg#0
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Norma Christine Presmeg is a retired mathematics education researcher whose work has concerned mathematical visualization, semiotics, and ethnomathematics, and their role in secondary-school mathematics teaching and learning. Presmeg is originally from South Africa, was educated in South Africa and England, and worked in the US, where she is professor emeritus of mathematics at Illinois State University. == Education and career == Presmeg has a bachelor's degree in mathematics and physics from Rhodes University in South Africa, with bachelor's honours in mathematics and a bachelor's degree in education from the University of Natal. She taught high school mathematics in South Africa from the mid-1960s to the late 1970s, and earned a master's degree in education in 1980 at the University of Natal. She completed a Ph.D. at the University of Cambridge in England, in 1985. Her dissertation, The Role of Visually Mediated Processes in High School Mathematics: A Classroom Investigation, was supervised by Alan J. Bishop. After returning to South Africa for a five-year stint at the University of Durban-Westville, she moved to the US in 1990, as a faculty member in Curriculum and Instruction at Florida State University. Ten years later, she moved to Illinois State University, where she is retired as a professor emeritus. == Selected publications == Presmeg's books include: Transitions Between Contexts of Mathematical Practices (edited with Guida de Abreu and Alan Bishop, Kluwer Academic Publishers, 2002) Critical Issues in Mathematics Education: Major Contributions of Alan Bishop (edited with Philip Clarkson, Springer, 2008) Approaches to Qualitative Research in Mathematics Education: Examples of Methodology and Methods (edited with Angelika Bikner-Ahsbahs and Christine Knipping, Springer, 2014) Semiotics in Mathematics Education (with Luis Radford, Wolff-Michael Roth, and Gert Kadunz, Springer, 2016) Signs of Signification: Semiotics in Mathematics Education Research (edited with Luis Radford, Wolff-Michael Roth, and Gert Kadunz, Springer, 2018) Compendium for Early Career Researchers in Mathematics Education (edited with Gabriele Kaiser, Springer, 2019) Her research and survey papers include: Presmeg, Norma C. (August 1986), "Visualisation and mathematical giftedness", Educational Studies in Mathematics, 17 (3): 297–311, doi:10.1007/bf00305075 Presmeg, Norma C. (November 1986), "Visualisation in high school mathematics", For the Learning of Mathematics, 6 (3): 42–46 Presmeg, Norma C. (December 1992), "Prototypes, metaphors, metonymies and imaginative rationality in high school mathematics", Educational Studies in Mathematics, 23 (6): 595–610, doi:10.1007/bf00540062 Aspinwall, Leslie; Shaw, Kenneth L.; Presmeg, Norma C. (1996), "Uncontrollable mental imagery: graphical connections between a function and its derivative", Educational Studies in Mathematics, 33 (3): 301–317, doi:10.1023/a:1002976729261 Presmeg, Norma C. (1998), "Ethnomathematics in teacher education", Journal of Mathematics Teacher Education, 1 (3): 317–339, doi:10.1023/a:1009946219294 Presmeg, Norma (January 2006), "Research on visualization in learning and teaching mathematics: emergence from psychology", Handbook of Research on the Psychology of Mathematics Education, BRILL, pp. 205–235, doi:10.1163/9789087901127_009, ISBN 9789077874660 Presmeg, Norma (February 2006), "Semiotics and the "connections" standard: significance of semiotics for teachers of mathematics", Educational Studies in Mathematics, 61 (1–2): 163–182, doi:10.1007/s10649-006-3365-z == References ==
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Wikipedia:Normal basis#0
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In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number theory, the study of the more refined question of the existence of a normal integral basis is part of Galois module theory. == Normal basis theorem == Let F ⊂ K {\displaystyle F\subset K} be a Galois extension with Galois group G {\displaystyle G} . The classical normal basis theorem states that there is an element β ∈ K {\displaystyle \beta \in K} such that { g ( β ) : g ∈ G } {\displaystyle \{g(\beta ):g\in G\}} forms a basis of K, considered as a vector space over F. That is, any element α ∈ K {\displaystyle \alpha \in K} can be written uniquely as α = ∑ g ∈ G a g g ( β ) {\textstyle \alpha =\sum _{g\in G}a_{g}\,g(\beta )} for some elements a g ∈ F . {\displaystyle a_{g}\in F.} A normal basis contrasts with a primitive element basis of the form { 1 , β , β 2 , … , β n − 1 } {\displaystyle \{1,\beta ,\beta ^{2},\ldots ,\beta ^{n-1}\}} , where β ∈ K {\displaystyle \beta \in K} is an element whose minimal polynomial has degree n = [ K : F ] {\displaystyle n=[K:F]} . == Group representation point of view == A field extension K / F with Galois group G can be naturally viewed as a representation of the group G over the field F in which each automorphism is represented by itself. Representations of G over the field F can be viewed as left modules for the group algebra F[G]. Every homomorphism of left F[G]-modules ϕ : F [ G ] → K {\displaystyle \phi :F[G]\rightarrow K} is of form ϕ ( r ) = r β {\displaystyle \phi (r)=r\beta } for some β ∈ K {\displaystyle \beta \in K} . Since { 1 ⋅ σ | σ ∈ G } {\displaystyle \{1\cdot \sigma |\sigma \in G\}} is a linear basis of F[G] over F, it follows easily that ϕ {\displaystyle \phi } is bijective iff β {\displaystyle \beta } generates a normal basis of K over F. The normal basis theorem therefore amounts to the statement saying that if K / F is finite Galois extension, then K ≅ F [ G ] {\displaystyle K\cong F[G]} as a left F [ G ] {\displaystyle F[G]} -module. In terms of representations of G over F, this means that K is isomorphic to the regular representation. == Case of finite fields == For finite fields this can be stated as follows: Let F = G F ( q ) = F q {\displaystyle F=\mathrm {GF} (q)=\mathbb {F} _{q}} denote the field of q elements, where q = pm is a prime power, and let K = G F ( q n ) = F q n {\displaystyle K=\mathrm {GF} (q^{n})=\mathbb {F} _{q^{n}}} denote its extension field of degree n ≥ 1. Here the Galois group is G = Gal ( K / F ) = { 1 , Φ , Φ 2 , … , Φ n − 1 } {\displaystyle G={\text{Gal}}(K/F)=\{1,\Phi ,\Phi ^{2},\ldots ,\Phi ^{n-1}\}} with Φ n = 1 , {\displaystyle \Phi ^{n}=1,} a cyclic group generated by the q-power Frobenius automorphism Φ ( α ) = α q , {\displaystyle \Phi (\alpha )=\alpha ^{q},} with Φ n = 1 = I d K . {\displaystyle \Phi ^{n}=1=\mathrm {Id} _{K}.} Then there exists an element β ∈ K such that { β , Φ ( β ) , Φ 2 ( β ) , … , Φ n − 1 ( β ) } = { β , β q , β q 2 , … , β q n − 1 } {\displaystyle \{\beta ,\Phi (\beta ),\Phi ^{2}(\beta ),\ldots ,\Phi ^{n-1}(\beta )\}\ =\ \{\beta ,\beta ^{q},\beta ^{q^{2}},\ldots ,\beta ^{q^{n-1}}\!\}} is a basis of K over F. === Proof for finite fields === In case the Galois group is cyclic as above, generated by Φ {\displaystyle \Phi } with Φ n = 1 , {\displaystyle \Phi ^{n}=1,} the normal basis theorem follows from two basic facts. The first is the linear independence of characters: a multiplicative character is a mapping χ from a group H to a field K satisfying χ ( h 1 h 2 ) = χ ( h 1 ) χ ( h 2 ) {\displaystyle \chi (h_{1}h_{2})=\chi (h_{1})\chi (h_{2})} ; then any distinct characters χ 1 , χ 2 , … {\displaystyle \chi _{1},\chi _{2},\ldots } are linearly independent in the K-vector space of mappings. We apply this to the Galois group automorphisms χ i = Φ i : K → K , {\displaystyle \chi _{i}=\Phi ^{i}:K\to K,} thought of as mappings from the multiplicative group H = K × {\displaystyle H=K^{\times }} . Now K ≅ F n {\displaystyle K\cong F^{n}} as an F-vector space, so we may consider Φ : F n → F n {\displaystyle \Phi :F^{n}\to F^{n}} as an element of the matrix algebra Mn(F); since its powers 1 , Φ , … , Φ n − 1 {\displaystyle 1,\Phi ,\ldots ,\Phi ^{n-1}} are linearly independent (over K and a fortiori over F), its minimal polynomial must have degree at least n, i.e. it must be X n − 1 {\displaystyle X^{n}-1} . The second basic fact is the classification of finitely generated modules over a PID such as F [ X ] {\displaystyle F[X]} . Every such module M can be represented as M ≅ ⨁ i = 1 k F [ X ] ( f i ( X ) ) {\textstyle M\cong \bigoplus _{i=1}^{k}{\frac {F[X]}{(f_{i}(X))}}} , where f i ( X ) {\displaystyle f_{i}(X)} may be chosen so that they are monic polynomials or zero and f i + 1 ( X ) {\displaystyle f_{i+1}(X)} is a multiple of f i ( X ) {\displaystyle f_{i}(X)} . f k ( X ) {\displaystyle f_{k}(X)} is the monic polynomial of smallest degree annihilating the module, or zero if no such non-zero polynomial exists. In the first case dim F M = ∑ i = 1 k deg f i {\textstyle \dim _{F}M=\sum _{i=1}^{k}\deg f_{i}} , in the second case dim F M = ∞ {\displaystyle \dim _{F}M=\infty } . In our case of cyclic G of size n generated by Φ {\displaystyle \Phi } we have an F-algebra isomorphism F [ G ] ≅ F [ X ] ( X n − 1 ) {\textstyle F[G]\cong {\frac {F[X]}{(X^{n}-1)}}} where X corresponds to 1 ⋅ Φ {\displaystyle 1\cdot \Phi } , so every F [ G ] {\displaystyle F[G]} -module may be viewed as an F [ X ] {\displaystyle F[X]} -module with multiplication by X being multiplication by 1 ⋅ Φ {\displaystyle 1\cdot \Phi } . In case of K this means X α = Φ ( α ) {\displaystyle X\alpha =\Phi (\alpha )} , so the monic polynomial of smallest degree annihilating K is the minimal polynomial of Φ {\displaystyle \Phi } . Since K is a finite dimensional F-space, the representation above is possible with f k ( X ) = X n − 1 {\displaystyle f_{k}(X)=X^{n}-1} . Since dim F ( K ) = n , {\displaystyle \dim _{F}(K)=n,} we can only have k = 1 {\displaystyle k=1} , and K ≅ F [ X ] ( X n − 1 ) {\textstyle K\cong {\frac {F[X]}{(X^{n}{-}\,1)}}} as F[X]-modules. (Note this is an isomorphism of F-linear spaces, but not of rings or F-algebras.) This gives isomorphism of F [ G ] {\displaystyle F[G]} -modules K ≅ F [ G ] {\displaystyle K\cong F[G]} that we talked about above, and under it the basis { 1 , X , X 2 , … , X n − 1 } {\displaystyle \{1,X,X^{2},\ldots ,X^{n-1}\}} on the right side corresponds to a normal basis { β , Φ ( β ) , Φ 2 ( β ) , … , Φ n − 1 ( β ) } {\displaystyle \{\beta ,\Phi (\beta ),\Phi ^{2}(\beta ),\ldots ,\Phi ^{n-1}(\beta )\}} of K on the left. Note that this proof would also apply in the case of a cyclic Kummer extension. === Example === Consider the field K = G F ( 2 3 ) = F 8 {\displaystyle K=\mathrm {GF} (2^{3})=\mathbb {F} _{8}} over F = G F ( 2 ) = F 2 {\displaystyle F=\mathrm {GF} (2)=\mathbb {F} _{2}} , with Frobenius automorphism Φ ( α ) = α 2 {\displaystyle \Phi (\alpha )=\alpha ^{2}} . The proof above clarifies the choice of normal bases in terms of the structure of K as a representation of G (or F[G]-module). The irreducible factorization X n − 1 = X 3 − 1 = ( X − 1 ) ( X 2 + X + 1 ) ∈ F [ X ] {\displaystyle X^{n}-1\ =\ X^{3}-1\ =\ (X{-}1)(X^{2}{+}X{+}1)\ \in \ F[X]} means we have a direct sum of F[G]-modules (by the Chinese remainder theorem): K ≅ F [ X ] ( X 3 − 1 ) ≅ F [ X ] ( X + 1 ) ⊕ F [ X ] ( X 2 + X + 1 ) . {\displaystyle K\ \cong \ {\frac {F[X]}{(X^{3}{-}\,1)}}\ \cong \ {\frac {F[X]}{(X{+}1)}}\oplus {\frac {F[X]}{(X^{2}{+}X{+}1)}}.} The first component is just F ⊂ K {\displaystyle F\subset K} , while the second is isomorphic as an F[G]-module to F 2 2 ≅ F 2 [ X ] / ( X 2 + X + 1 ) {\displaystyle \mathbb {F} _{2^{2}}\cong \mathbb {F} _{2}[X]/(X^{2}{+}X{+}1)} under the action Φ ⋅ X i = X i + 1 . {\displaystyle \Phi \cdot X^{i}=X^{i+1}.} (Thus K ≅ F 2 ⊕ F 4 {\displaystyle K\cong \mathbb {F} _{2}\oplus \mathbb {F} _{4}} as F[G]-modules, but not as F-algebras.) The elements β ∈ K {\displaystyle \beta \in K} which can be used for a normal basis are precisely those outside either of the submodules, so that ( Φ + 1 ) ( β ) ≠ 0 {\displaystyle (\Phi {+}1)(\beta )\neq 0} and ( Φ 2 + Φ + 1 ) ( β ) ≠ 0 {\displaystyle (\Phi ^{2}{+}\Phi {+}1)(\beta )\neq 0} . In terms of the G-orbits of K, which correspond to the irreducible factors of: t 2 3 − t = t ( t + 1 ) ( t 3 + t + 1 ) ( t 3 + t 2 + 1 ) ∈ F [ t ] , {\displaystyle t^{2^{3}}-t\ =\ t(t{+}1)\left(t^{3}+t+1\right)\left(t^{3}+t^{2}+1\right)\ \in \ F[t],} the elements of F = F 2 {\displaystyle F=\mathbb {F} _{2}} are the roots of t ( t + 1 ) {\displaystyle t(t{+}1)} , the nonzero elements of the submodule F 4 {\displaystyle \mathbb {F} _{4}} are the roots of t 3 + t + 1 {\displaystyle t^{3}+t+1} , while the normal basis, which in this case is unique, is given by the roots of the remaining factor t 3 + t 2 + 1 {\displaystyle t^{3}{+}t^{2}{+}1} . By contrast, for the extension field L = G F ( 2 4 ) = F 16 {\displaystyle L=\mathrm {GF} (2^{4})=\mathbb {F} _{16}} in which n = 4 is divisible by p = 2, we have the F[G]-module isomorphism L ≅ F 2 [ X ] / ( X 4 − 1 ) = F 2 [ X ] / ( X + 1 ) 4 . {\displaystyle L\ \cong \ \mathbb {F} _{2}[X]/(X^{4}{-}1)\ =\ \mathbb {F} _{2}[X]/(X{+}1)^{4}.} Here the operator Φ ≅ X {\displaystyle \Phi \cong X} is not diagonalizable, the module L has nested submodules given by generalized eigenspaces of Φ {\displaystyle \Phi } , and the normal basis elements β are those outside the largest proper generalized eigenspace, the elements with ( Φ + 1 ) 3 ( β ) ≠ 0 {\displaystyle (\Phi {+}1)^{3}(\beta )\neq 0} . === Application to cryptography === The normal basis is frequently used in cryptographic applications based on the discrete logarithm problem, such as elliptic curve cryptography, since arithmetic using a normal basis is typically more computationally efficient than using other bases. For example, in the field K = G F ( 2 3 ) = F 8 {\displaystyle K=\mathrm {GF} (2^{3})=\mathbb {F} _{8}} above, we may represent elements as bit-strings: α = ( a 2 , a 1 , a 0 ) = a 2 Φ 2 ( β ) + a 1 Φ ( β ) + a 0 β = a 2 β 4 + a 1 β 2 + a 0 β , {\displaystyle \alpha \ =\ (a_{2},a_{1},a_{0})\ =\ a_{2}\Phi ^{2}(\beta )+a_{1}\Phi (\beta )+a_{0}\beta \ =\ a_{2}\beta ^{4}+a_{1}\beta ^{2}+a_{0}\beta ,} where the coefficients are bits a i ∈ G F ( 2 ) = { 0 , 1 } . {\displaystyle a_{i}\in \mathrm {GF} (2)=\{0,1\}.} Now we can square elements by doing a left circular shift, α 2 = Φ ( a 2 , a 1 , a 0 ) = ( a 1 , a 0 , a 2 ) {\displaystyle \alpha ^{2}=\Phi (a_{2},a_{1},a_{0})=(a_{1},a_{0},a_{2})} , since squaring β4 gives β8 = β. This makes the normal basis especially attractive for cryptosystems that utilize frequent squaring. == Proof for the case of infinite fields == Suppose K / F {\displaystyle K/F} is a finite Galois extension of the infinite field F. Let [K : F] = n, Gal ( K / F ) = G = { σ 1 . . . σ n } {\displaystyle {\text{Gal}}(K/F)=G=\{\sigma _{1}...\sigma _{n}\}} , where σ 1 = Id {\displaystyle \sigma _{1}={\text{Id}}} . By the primitive element theorem there exists α ∈ K {\displaystyle \alpha \in K} such i ≠ j ⟹ σ i ( α ) ≠ σ j ( α ) {\displaystyle i\neq j\implies \sigma _{i}(\alpha )\neq \sigma _{j}(\alpha )} and K = F [ α ] {\displaystyle K=F[\alpha ]} . Let us write α i = σ i ( α ) {\displaystyle \alpha _{i}=\sigma _{i}(\alpha )} . α {\displaystyle \alpha } 's (monic) minimal polynomial f over K is the irreducible degree n polynomial given by the formula f ( X ) = ∏ i = 1 n ( X − α i ) {\displaystyle {\begin{aligned}f(X)&=\prod _{i=1}^{n}(X-\alpha _{i})\end{aligned}}} Since f is separable (it has simple roots) we may define g ( X ) = f ( X ) ( X − α ) f ′ ( α ) g i ( X ) = f ( X ) ( X − α i ) f ′ ( α i ) = σ i ( g ( X ) ) . {\displaystyle {\begin{aligned}g(X)&=\ {\frac {f(X)}{(X-\alpha )f'(\alpha )}}\\g_{i}(X)&=\ {\frac {f(X)}{(X-\alpha _{i})f'(\alpha _{i})}}=\ \sigma _{i}(g(X)).\end{aligned}}} In other words, g i ( X ) = ∏ 1 ≤ j ≤ n j ≠ i X − α j α i − α j g ( X ) = g 1 ( X ) . {\displaystyle {\begin{aligned}g_{i}(X)&=\prod _{\begin{array}{c}1\leq j\leq n\\j\neq i\end{array}}{\frac {X-\alpha _{j}}{\alpha _{i}-\alpha _{j}}}\\g(X)&=g_{1}(X).\end{aligned}}} Note that g ( α ) = 1 {\displaystyle g(\alpha )=1} and g i ( α ) = 0 {\displaystyle g_{i}(\alpha )=0} for i ≠ 1 {\displaystyle i\neq 1} . Next, define an n × n {\displaystyle n\times n} matrix A of polynomials over K and a polynomial D by A i j ( X ) = σ i ( σ j ( g ( X ) ) = σ i ( g j ( X ) ) D ( X ) = det A ( X ) . {\displaystyle {\begin{aligned}A_{ij}(X)&=\sigma _{i}(\sigma _{j}(g(X))=\sigma _{i}(g_{j}(X))\\D(X)&=\det A(X).\end{aligned}}} Observe that A i j ( X ) = g k ( X ) {\displaystyle A_{ij}(X)=g_{k}(X)} , where k is determined by σ k = σ i ⋅ σ j {\displaystyle \sigma _{k}=\sigma _{i}\cdot \sigma _{j}} ; in particular k = 1 {\displaystyle k=1} iff σ i = σ j − 1 {\displaystyle \sigma _{i}=\sigma _{j}^{-1}} . It follows that A ( α ) {\displaystyle A(\alpha )} is the permutation matrix corresponding to the permutation of G which sends each σ i {\displaystyle \sigma _{i}} to σ i − 1 {\displaystyle \sigma _{i}^{-1}} . (We denote by A ( α ) {\displaystyle A(\alpha )} the matrix obtained by evaluating A ( X ) {\displaystyle A(X)} at x = α {\displaystyle x=\alpha } .) Therefore, D ( α ) = det A ( α ) = ± 1 {\displaystyle D(\alpha )=\det A(\alpha )=\pm 1} . We see that D is a non-zero polynomial, and therefore it has only a finite number of roots. Since we assumed F is infinite, we can find a ∈ F {\displaystyle a\in F} such that D ( a ) ≠ 0 {\displaystyle D(a)\neq 0} . Define β = g ( a ) β i = g i ( a ) = σ i ( β ) . {\displaystyle {\begin{aligned}\beta &=g(a)\\\beta _{i}&=g_{i}(a)=\sigma _{i}(\beta ).\end{aligned}}} We claim that { β 1 , … , β n } {\displaystyle \{\beta _{1},\ldots ,\beta _{n}\}} is a normal basis. We only have to show that β 1 , … , β n {\displaystyle \beta _{1},\ldots ,\beta _{n}} are linearly independent over F, so suppose ∑ j = 1 n x j β j = 0 {\textstyle \sum _{j=1}^{n}x_{j}\beta _{j}=0} for some x 1 . . . x n ∈ F {\displaystyle x_{1}...x_{n}\in F} . Applying the automorphism σ i {\displaystyle \sigma _{i}} yields ∑ j = 1 n x j σ i ( g j ( a ) ) = 0 {\textstyle \sum _{j=1}^{n}x_{j}\sigma _{i}(g_{j}(a))=0} for all i. In other words, A ( a ) ⋅ x ¯ = 0 ¯ {\displaystyle A(a)\cdot {\overline {x}}={\overline {0}}} . Since det A ( a ) = D ( a ) ≠ 0 {\displaystyle \det A(a)=D(a)\neq 0} , we conclude that x ¯ = 0 ¯ {\displaystyle {\overline {x}}={\overline {0}}} , which completes the proof. It is tempting to take a = α {\displaystyle a=\alpha } because D ( α ) ≠ 0 {\displaystyle D(\alpha )\neq 0} . But this is impermissible because we used the fact that a ∈ F {\displaystyle a\in F} to conclude that for any F-automorphism σ {\displaystyle \sigma } and polynomial h ( X ) {\displaystyle h(X)} over K {\displaystyle K} the value of the polynomial σ ( h ( X ) ) {\displaystyle \sigma (h(X))} at a equals σ ( h ( a ) ) {\displaystyle \sigma (h(a))} . == Primitive normal basis == A primitive normal basis of an extension of finite fields E / F is a normal basis for E / F that is generated by a primitive element of E, that is a generator of the multiplicative group K×. (Note that this is a more restrictive definition of primitive element than that mentioned above after the general normal basis theorem: one requires powers of the element to produce every non-zero element of K, not merely a basis.) Lenstra and Schoof (1987) proved that every extension of finite fields possesses a primitive normal basis, the case when F is a prime field having been settled by Harold Davenport. == Free elements == If K / F is a Galois extension and x in K generates a normal basis over F, then x is free in K / F. If x has the property that for every subgroup H of the Galois group G, with fixed field KH, x is free for K / KH, then x is said to be completely free in K / F. Every Galois extension has a completely free element. == See also == Dual basis in a field extension Polynomial basis Zech's logarithm == References == Cohen, S.; Niederreiter, H., eds. (1996). Finite Fields and Applications. Proceedings of the 3rd international conference, Glasgow, UK, July 11–14, 1995. London Mathematical Society Lecture Note Series. Vol. 233. Cambridge University Press. ISBN 978-0-521-56736-7. Zbl 0851.00052. Lenstra, H.W. Jr; Schoof, R.J. (1987). "Primitive normal bases for finite fields". Mathematics of Computation. 48 (177): 217–231. doi:10.2307/2007886. hdl:1887/3824. JSTOR 2007886. Zbl 0615.12023. Menezes, Alfred J., ed. (1993). Applications of finite fields. The Kluwer International Series in Engineering and Computer Science. Vol. 199. Boston: Kluwer Academic Publishers. ISBN 978-0792392828. Zbl 0779.11059.
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Wikipedia:Normal convergence#0
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In mathematics normal convergence is a type of convergence for series of functions. Like absolute-convergence, it has the useful property that it is preserved when the order of summation is changed. == History == The concept of normal convergence was first introduced by René Baire in 1908 in his book Leçons sur les théories générales de l'analyse. == Definition == Given a set S and functions f n : S → C {\displaystyle f_{n}:S\to \mathbb {C} } (or to any normed vector space), the series ∑ n = 0 ∞ f n ( x ) {\displaystyle \sum _{n=0}^{\infty }f_{n}(x)} is called normally convergent if the series of uniform norms of the terms of the series converges, i.e., ∑ n = 0 ∞ ‖ f n ‖ := ∑ n = 0 ∞ sup x ∈ S | f n ( x ) | < ∞ . {\displaystyle \sum _{n=0}^{\infty }\|f_{n}\|:=\sum _{n=0}^{\infty }\sup _{x\in S}|f_{n}(x)|<\infty .} == Distinctions == Normal convergence implies uniform absolute convergence, i.e., uniform convergence of the series of nonnegative functions ∑ n = 0 ∞ | f n ( x ) | {\displaystyle \sum _{n=0}^{\infty }|f_{n}(x)|} ; this fact is essentially the Weierstrass M-test. However, they should not be confused; to illustrate this, consider f n ( x ) = { 1 / n , x = n , 0 , x ≠ n . {\displaystyle f_{n}(x)={\begin{cases}1/n,&x=n,\\0,&x\neq n.\end{cases}}} Then the series ∑ n = 0 ∞ | f n ( x ) | {\displaystyle \sum _{n=0}^{\infty }|f_{n}(x)|} is uniformly convergent (for any ε take n ≥ 1/ε), but the series of uniform norms is the harmonic series and thus diverges. An example using continuous functions can be made by replacing these functions with bump functions of height 1/n and width 1 centered at each natural number n. As well, normal convergence of a series is different from norm-topology convergence, i.e. convergence of the partial sum sequence in the topology induced by the uniform norm. Normal convergence implies norm-topology convergence if and only if the space of functions under consideration is complete with respect to the uniform norm. (The converse does not hold even for complete function spaces: for example, consider the harmonic series as a sequence of constant functions). == Generalizations == === Local normal convergence === A series can be called "locally normally convergent on X" if each point x in X has a neighborhood U such that the series of functions ƒn restricted to the domain U ∑ n = 0 ∞ f n ∣ U {\displaystyle \sum _{n=0}^{\infty }f_{n}\mid _{U}} is normally convergent, i.e. such that ∑ n = 0 ∞ ‖ f n ‖ U < ∞ {\displaystyle \sum _{n=0}^{\infty }\|f_{n}\|_{U}<\infty } where the norm ‖ ⋅ ‖ U {\displaystyle \|\cdot \|_{U}} is the supremum over the domain U. === Compact normal convergence === A series is said to be "normally convergent on compact subsets of X" or "compactly normally convergent on X" if for every compact subset K of X, the series of functions ƒn restricted to K ∑ n = 0 ∞ f n ∣ K {\displaystyle \sum _{n=0}^{\infty }f_{n}\mid _{K}} is normally convergent on K. Note: if X is locally compact (even in the weakest sense), local normal convergence and compact normal convergence are equivalent. == Properties == Every normal convergent series is uniformly convergent, locally uniformly convergent, and compactly uniformly convergent. This is very important, since it assures that any re-arrangement of the series, any derivatives or integrals of the series, and sums and products with other convergent series will converge to the "correct" value. If ∑ n = 0 ∞ f n ( x ) {\displaystyle \sum _{n=0}^{\infty }f_{n}(x)} is normally convergent to f {\displaystyle f} , then any re-arrangement of the sequence (ƒ1, ƒ2, ƒ3 ...) also converges normally to the same ƒ. That is, for every bijection τ : N → N {\displaystyle \tau :\mathbb {N} \to \mathbb {N} } , ∑ n = 0 ∞ f τ ( n ) ( x ) {\displaystyle \sum _{n=0}^{\infty }f_{\tau (n)}(x)} is normally convergent to f {\displaystyle f} . == See also == Modes of convergence (annotated index) == References ==
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Wikipedia:Normal element#0
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In mathematics, an element of a *-algebra is called normal if it commutates with its adjoint. == Definition == Let A {\displaystyle {\mathcal {A}}} be a *-Algebra. An element a ∈ A {\displaystyle a\in {\mathcal {A}}} is called normal if it commutes with a ∗ {\displaystyle a^{*}} , i.e. it satisfies the equation a a ∗ = a ∗ a {\displaystyle aa^{*}=a^{*}a} . The set of normal elements is denoted by A N {\displaystyle {\mathcal {A}}_{N}} or N ( A ) {\displaystyle N({\mathcal {A}})} . A special case of particular importance is the case where A {\displaystyle {\mathcal {A}}} is a complete normed *-algebra, that satisfies the C*-identity ( ‖ a ∗ a ‖ = ‖ a ‖ 2 ∀ a ∈ A {\displaystyle \left\|a^{*}a\right\|=\left\|a\right\|^{2}\ \forall a\in {\mathcal {A}}} ), which is called a C*-algebra. == Examples == Every self-adjoint element of a a *-algebra is normal. Every unitary element of a a *-algebra is normal. If A {\displaystyle {\mathcal {A}}} is a C*-Algebra and a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} a normal element, then for every continuous function f {\displaystyle f} on the spectrum of a {\displaystyle a} the continuous functional calculus defines another normal element f ( a ) {\displaystyle f(a)} . == Criteria == Let A {\displaystyle {\mathcal {A}}} be a *-algebra. Then: An element a ∈ A {\displaystyle a\in {\mathcal {A}}} is normal if and only if the *-subalgebra generated by a {\displaystyle a} , meaning the smallest *-algebra containing a {\displaystyle a} , is commutative. Every element a ∈ A {\displaystyle a\in {\mathcal {A}}} can be uniquely decomposed into a real and imaginary part, which means there exist self-adjoint elements a 1 , a 2 ∈ A s a {\displaystyle a_{1},a_{2}\in {\mathcal {A}}_{sa}} , such that a = a 1 + i a 2 {\displaystyle a=a_{1}+\mathrm {i} a_{2}} , where i {\displaystyle \mathrm {i} } denotes the imaginary unit. Exactly then a {\displaystyle a} is normal if a 1 a 2 = a 2 a 1 {\displaystyle a_{1}a_{2}=a_{2}a_{1}} , i.e. real and imaginary part commutate. == Properties == === In *-algebras === Let a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} be a normal element of a *-algebra A {\displaystyle {\mathcal {A}}} . Then: The adjoint element a ∗ {\displaystyle a^{*}} is also normal, since a = ( a ∗ ) ∗ {\displaystyle a=(a^{*})^{*}} holds for the involution *. === In C*-algebras === Let a ∈ A N {\displaystyle a\in {\mathcal {A}}_{N}} be a normal element of a C*-algebra A {\displaystyle {\mathcal {A}}} . Then: It is ‖ a 2 ‖ = ‖ a ‖ 2 {\displaystyle \left\|a^{2}\right\|=\left\|a\right\|^{2}} , since for normal elements using the C*-identity ‖ a 2 ‖ 2 = ‖ ( a 2 ) ( a 2 ) ∗ ‖ = ‖ ( a ∗ a ) ∗ ( a ∗ a ) ‖ = ‖ a ∗ a ‖ 2 = ( ‖ a ‖ 2 ) 2 {\displaystyle \left\|a^{2}\right\|^{2}=\left\|(a^{2})(a^{2})^{*}\right\|=\left\|(a^{*}a)^{*}(a^{*}a)\right\|=\left\|a^{*}a\right\|^{2}=\left(\left\|a\right\|^{2}\right)^{2}} holds. Every normal element is a normaloid element, i.e. the spectral radius r ( a ) {\displaystyle r(a)} equals the norm of a {\displaystyle a} , i.e. r ( a ) = ‖ a ‖ {\displaystyle r(a)=\left\|a\right\|} . This follows from the spectral radius formula by repeated application of the previous property. A continuous functional calculus can be developed which – put simply – allows the application of continuous functions on the spectrum of a {\displaystyle a} to a {\displaystyle a} . == See also == Normal matrix Normal operator == Notes == == References == Dixmier, Jacques (1977). C*-algebras. Translated by Jellett, Francis. Amsterdam/New York/Oxford: North-Holland. ISBN 0-7204-0762-1. English translation of Les C*-algèbres et leurs représentations (in French). Gauthier-Villars. 1969. Heuser, Harro (1982). Functional analysis. Translated by Horvath, John. John Wiley & Sons Ltd. ISBN 0-471-10069-2. Werner, Dirk (2018). Funktionalanalysis (in German) (8 ed.). Springer. ISBN 978-3-662-55407-4.
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Wikipedia:Normal homomorphism#0
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In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N {\displaystyle N} of the group G {\displaystyle G} is normal in G {\displaystyle G} if and only if g n g − 1 ∈ N {\displaystyle gng^{-1}\in N} for all g ∈ G {\displaystyle g\in G} and n ∈ N {\displaystyle n\in N} . The usual notation for this relation is N ◃ G {\displaystyle N\triangleleft G} . Normal subgroups are important because they (and only they) can be used to construct quotient groups of the given group. Furthermore, the normal subgroups of G {\displaystyle G} are precisely the kernels of group homomorphisms with domain G {\displaystyle G} , which means that they can be used to internally classify those homomorphisms. Évariste Galois was the first to realize the importance of the existence of normal subgroups. == Definitions == A subgroup N {\displaystyle N} of a group G {\displaystyle G} is called a normal subgroup of G {\displaystyle G} if it is invariant under conjugation; that is, the conjugation of an element of N {\displaystyle N} by an element of G {\displaystyle G} is always in N {\displaystyle N} . The usual notation for this relation is N ◃ G {\displaystyle N\triangleleft G} . === Equivalent conditions === For any subgroup N {\displaystyle N} of G {\displaystyle G} , the following conditions are equivalent to N {\displaystyle N} being a normal subgroup of G {\displaystyle G} . Therefore, any one of them may be taken as the definition. The image of conjugation of N {\displaystyle N} by any element of G {\displaystyle G} is a subset of N {\displaystyle N} , i.e., g N g − 1 ⊆ N {\displaystyle gNg^{-1}\subseteq N} for all g ∈ G {\displaystyle g\in G} . The image of conjugation of N {\displaystyle N} by any element of G {\displaystyle G} is equal to N , {\displaystyle N,} i.e., g N g − 1 = N {\displaystyle gNg^{-1}=N} for all g ∈ G {\displaystyle g\in G} . For all g ∈ G {\displaystyle g\in G} , the left and right cosets g N {\displaystyle gN} and N g {\displaystyle Ng} are equal. The sets of left and right cosets of N {\displaystyle N} in G {\displaystyle G} coincide. Multiplication in G {\displaystyle G} preserves the equivalence relation "is in the same left coset as". That is, for every g , g ′ , h , h ′ ∈ G {\displaystyle g,g',h,h'\in G} satisfying g N = g ′ N {\displaystyle gN=g'N} and h N = h ′ N {\displaystyle hN=h'N} , we have ( g h ) N = ( g ′ h ′ ) N {\displaystyle (gh)N=(g'h')N} . There exists a group on the set of left cosets of N {\displaystyle N} where multiplication of any two left cosets g N {\displaystyle gN} and h N {\displaystyle hN} yields the left coset ( g h ) N {\displaystyle (gh)N} (this group is called the quotient group of G {\displaystyle G} modulo N {\displaystyle N} , denoted G / N {\displaystyle G/N} ). N {\displaystyle N} is a union of conjugacy classes of G {\displaystyle G} . N {\displaystyle N} is preserved by the inner automorphisms of G {\displaystyle G} . There is some group homomorphism G → H {\displaystyle G\to H} whose kernel is N {\displaystyle N} . There exists a group homomorphism ϕ : G → H {\displaystyle \phi :G\to H} whose fibers form a group where the identity element is N {\displaystyle N} and multiplication of any two fibers ϕ − 1 ( h 1 ) {\displaystyle \phi ^{-1}(h_{1})} and ϕ − 1 ( h 2 ) {\displaystyle \phi ^{-1}(h_{2})} yields the fiber ϕ − 1 ( h 1 h 2 ) {\displaystyle \phi ^{-1}(h_{1}h_{2})} (this group is the same group G / N {\displaystyle G/N} mentioned above). There is some congruence relation on G {\displaystyle G} for which the equivalence class of the identity element is N {\displaystyle N} . For all n ∈ N {\displaystyle n\in N} and g ∈ G {\displaystyle g\in G} . the commutator [ n , g ] = n − 1 g − 1 n g {\displaystyle [n,g]=n^{-1}g^{-1}ng} is in N {\displaystyle N} . Any two elements commute modulo the normal subgroup membership relation. That is, for all g , h ∈ G {\displaystyle g,h\in G} , g h ∈ N {\displaystyle gh\in N} if and only if h g ∈ N {\displaystyle hg\in N} . == Examples == For any group G {\displaystyle G} , the trivial subgroup { e } {\displaystyle \{e\}} consisting of only the identity element of G {\displaystyle G} is always a normal subgroup of G {\displaystyle G} . Likewise, G {\displaystyle G} itself is always a normal subgroup of G {\displaystyle G} (if these are the only normal subgroups, then G {\displaystyle G} is said to be simple). Other named normal subgroups of an arbitrary group include the center of the group (the set of elements that commute with all other elements) and the commutator subgroup [ G , G ] {\displaystyle [G,G]} . More generally, since conjugation is an isomorphism, any characteristic subgroup is a normal subgroup. If G {\displaystyle G} is an abelian group then every subgroup N {\displaystyle N} of G {\displaystyle G} is normal, because g N = { g n } n ∈ N = { n g } n ∈ N = N g {\displaystyle gN=\{gn\}_{n\in N}=\{ng\}_{n\in N}=Ng} . More generally, for any group G {\displaystyle G} , every subgroup of the center Z ( G ) {\displaystyle Z(G)} of G {\displaystyle G} is normal in G {\displaystyle G} (in the special case that G {\displaystyle G} is abelian, the center is all of G {\displaystyle G} , hence the fact that all subgroups of an abelian group are normal). A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group. A concrete example of a normal subgroup is the subgroup N = { ( 1 ) , ( 123 ) , ( 132 ) } {\displaystyle N=\{(1),(123),(132)\}} of the symmetric group S 3 {\displaystyle S_{3}} , consisting of the identity and both three-cycles. In particular, one can check that every coset of N {\displaystyle N} is either equal to N {\displaystyle N} itself or is equal to ( 12 ) N = { ( 12 ) , ( 23 ) , ( 13 ) } {\displaystyle (12)N=\{(12),(23),(13)\}} . On the other hand, the subgroup H = { ( 1 ) , ( 12 ) } {\displaystyle H=\{(1),(12)\}} is not normal in S 3 {\displaystyle S_{3}} since ( 123 ) H = { ( 123 ) , ( 13 ) } ≠ { ( 123 ) , ( 23 ) } = H ( 123 ) {\displaystyle (123)H=\{(123),(13)\}\neq \{(123),(23)\}=H(123)} . This illustrates the general fact that any subgroup H ≤ G {\displaystyle H\leq G} of index two is normal. As an example of a normal subgroup within a matrix group, consider the general linear group G L n ( R ) {\displaystyle \mathrm {GL} _{n}(\mathbf {R} )} of all invertible n × n {\displaystyle n\times n} matrices with real entries under the operation of matrix multiplication and its subgroup S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )} of all n × n {\displaystyle n\times n} matrices of determinant 1 (the special linear group). To see why the subgroup S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )} is normal in G L n ( R ) {\displaystyle \mathrm {GL} _{n}(\mathbf {R} )} , consider any matrix X {\displaystyle X} in S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )} and any invertible matrix A {\displaystyle A} . Then using the two important identities det ( A B ) = det ( A ) det ( B ) {\displaystyle \det(AB)=\det(A)\det(B)} and det ( A − 1 ) = det ( A ) − 1 {\displaystyle \det(A^{-1})=\det(A)^{-1}} , one has that det ( A X A − 1 ) = det ( A ) det ( X ) det ( A ) − 1 = det ( X ) = 1 {\displaystyle \det(AXA^{-1})=\det(A)\det(X)\det(A)^{-1}=\det(X)=1} , and so A X A − 1 ∈ S L n ( R ) {\displaystyle AXA^{-1}\in \mathrm {SL} _{n}(\mathbf {R} )} as well. This means S L n ( R ) {\displaystyle \mathrm {SL} _{n}(\mathbf {R} )} is closed under conjugation in G L n ( R ) {\displaystyle \mathrm {GL} _{n}(\mathbf {R} )} , so it is a normal subgroup. In the Rubik's Cube group, the subgroups consisting of operations which only affect the orientations of either the corner pieces or the edge pieces are normal. The translation group is a normal subgroup of the Euclidean group in any dimension. This means: applying a rigid transformation, followed by a translation and then the inverse rigid transformation, has the same effect as a single translation. By contrast, the subgroup of all rotations about the origin is not a normal subgroup of the Euclidean group, as long as the dimension is at least 2: first translating, then rotating about the origin, and then translating back will typically not fix the origin and will therefore not have the same effect as a single rotation about the origin. == Properties == If H {\displaystyle H} is a normal subgroup of G {\displaystyle G} , and K {\displaystyle K} is a subgroup of G {\displaystyle G} containing H {\displaystyle H} , then H {\displaystyle H} is a normal subgroup of K {\displaystyle K} . A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. The smallest group exhibiting this phenomenon is the dihedral group of order 8. However, a characteristic subgroup of a normal subgroup is normal. A group in which normality is transitive is called a T-group. The two groups G {\displaystyle G} and H {\displaystyle H} are normal subgroups of their direct product G × H {\displaystyle G\times H} . If the group G {\displaystyle G} is a semidirect product G = N ⋊ H {\displaystyle G=N\rtimes H} , then N {\displaystyle N} is normal in G {\displaystyle G} , though H {\displaystyle H} need not be normal in G {\displaystyle G} . If M {\displaystyle M} and N {\displaystyle N} are normal subgroups of an additive group G {\displaystyle G} such that G = M + N {\displaystyle G=M+N} and M ∩ N = { 0 } {\displaystyle M\cap N=\{0\}} , then G = M ⊕ N {\displaystyle G=M\oplus N} . Normality is preserved under surjective homomorphisms; that is, if G → H {\displaystyle G\to H} is a surjective group homomorphism and N {\displaystyle N} is normal in G {\displaystyle G} , then the image f ( N ) {\displaystyle f(N)} is normal in H {\displaystyle H} . Normality is preserved by taking inverse images; that is, if G → H {\displaystyle G\to H} is a group homomorphism and N {\displaystyle N} is normal in H {\displaystyle H} , then the inverse image f − 1 ( N ) {\displaystyle f^{-1}(N)} is normal in G {\displaystyle G} . Normality is preserved on taking direct products; that is, if N 1 ◃ G 1 {\displaystyle N_{1}\triangleleft G_{1}} and N 2 ◃ G 2 {\displaystyle N_{2}\triangleleft G_{2}} , then N 1 × N 2 ◃ G 1 × G 2 {\displaystyle N_{1}\times N_{2}\;\triangleleft \;G_{1}\times G_{2}} . Every subgroup of index 2 is normal. More generally, a subgroup, H {\displaystyle H} , of finite index, n {\displaystyle n} , in G {\displaystyle G} contains a subgroup, K , {\displaystyle K,} normal in G {\displaystyle G} and of index dividing n ! {\displaystyle n!} called the normal core. In particular, if p {\displaystyle p} is the smallest prime dividing the order of G {\displaystyle G} , then every subgroup of index p {\displaystyle p} is normal. The fact that normal subgroups of G {\displaystyle G} are precisely the kernels of group homomorphisms defined on G {\displaystyle G} accounts for some of the importance of normal subgroups; they are a way to internally classify all homomorphisms defined on a group. For example, a non-identity finite group is simple if and only if it is isomorphic to all of its non-identity homomorphic images, a finite group is perfect if and only if it has no normal subgroups of prime index, and a group is imperfect if and only if the derived subgroup is not supplemented by any proper normal subgroup. === Lattice of normal subgroups === Given two normal subgroups, N {\displaystyle N} and M {\displaystyle M} , of G {\displaystyle G} , their intersection N ∩ M {\displaystyle N\cap M} and their product N M = { n m : n ∈ N and m ∈ M } {\displaystyle NM=\{nm:n\in N\;{\text{ and }}\;m\in M\}} are also normal subgroups of G {\displaystyle G} . The normal subgroups of G {\displaystyle G} form a lattice under subset inclusion with least element, { e } {\displaystyle \{e\}} , and greatest element, G {\displaystyle G} . The meet of two normal subgroups, N {\displaystyle N} and M {\displaystyle M} , in this lattice is their intersection and the join is their product. The lattice is complete and modular. == Normal subgroups, quotient groups and homomorphisms == If N {\displaystyle N} is a normal subgroup, we can define a multiplication on cosets as follows: ( a 1 N ) ( a 2 N ) := ( a 1 a 2 ) N {\displaystyle \left(a_{1}N\right)\left(a_{2}N\right):=\left(a_{1}a_{2}\right)N} This relation defines a mapping G / N × G / N → G / N {\displaystyle G/N\times G/N\to G/N} . To show that this mapping is well-defined, one needs to prove that the choice of representative elements a 1 , a 2 {\displaystyle a_{1},a_{2}} does not affect the result. To this end, consider some other representative elements a 1 ′ ∈ a 1 N , a 2 ′ ∈ a 2 N {\displaystyle a_{1}'\in a_{1}N,a_{2}'\in a_{2}N} . Then there are n 1 , n 2 ∈ N {\displaystyle n_{1},n_{2}\in N} such that a 1 ′ = a 1 n 1 , a 2 ′ = a 2 n 2 {\displaystyle a_{1}'=a_{1}n_{1},a_{2}'=a_{2}n_{2}} . It follows that a 1 ′ a 2 ′ N = a 1 n 1 a 2 n 2 N = a 1 a 2 n 1 ′ n 2 N = a 1 a 2 N {\displaystyle a_{1}'a_{2}'N=a_{1}n_{1}a_{2}n_{2}N=a_{1}a_{2}n_{1}'n_{2}N=a_{1}a_{2}N} where we also used the fact that N {\displaystyle N} is a normal subgroup, and therefore there is n 1 ′ ∈ N {\displaystyle n_{1}'\in N} such that n 1 a 2 = a 2 n 1 ′ {\displaystyle n_{1}a_{2}=a_{2}n_{1}'} . This proves that this product is a well-defined mapping between cosets. With this operation, the set of cosets is itself a group, called the quotient group and denoted with G / N . {\displaystyle G/N.} There is a natural homomorphism, f : G → G / N {\displaystyle f:G\to G/N} , given by f ( a ) = a N {\displaystyle f(a)=aN} . This homomorphism maps N {\displaystyle N} into the identity element of G / N {\displaystyle G/N} , which is the coset e N = N {\displaystyle eN=N} , that is, ker ( f ) = N {\displaystyle \ker(f)=N} . In general, a group homomorphism, f : G → H {\displaystyle f:G\to H} sends subgroups of G {\displaystyle G} to subgroups of H {\displaystyle H} . Also, the preimage of any subgroup of H {\displaystyle H} is a subgroup of G {\displaystyle G} . We call the preimage of the trivial group { e } {\displaystyle \{e\}} in H {\displaystyle H} the kernel of the homomorphism and denote it by ker f {\displaystyle \ker f} . As it turns out, the kernel is always normal and the image of G , f ( G ) {\displaystyle G,f(G)} , is always isomorphic to G / ker f {\displaystyle G/\ker f} (the first isomorphism theorem). In fact, this correspondence is a bijection between the set of all quotient groups of G {\displaystyle G} , G / N {\displaystyle G/N} , and the set of all homomorphic images of G {\displaystyle G} (up to isomorphism). It is also easy to see that the kernel of the quotient map, f : G → G / N {\displaystyle f:G\to G/N} , is N {\displaystyle N} itself, so the normal subgroups are precisely the kernels of homomorphisms with domain G {\displaystyle G} . == See also == == Notes == == References == == Bibliography == == Further reading == I. N. Herstein, Topics in algebra. Second edition. Xerox College Publishing, Lexington, Mass.-Toronto, Ont., 1975. xi+388 pp. == External links == Weisstein, Eric W. "normal subgroup". MathWorld. Normal subgroup in Springer's Encyclopedia of Mathematics Robert Ash: Group Fundamentals in Abstract Algebra. The Basic Graduate Year Timothy Gowers, Normal subgroups and quotient groups John Baez, What's a Normal Subgroup?
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Wikipedia:Norman Fenton#0
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Norman Elliott Fenton (born 18 May 1956) is a British mathematician and computer scientist. He is the Professor of Risk Information Management in the School of Electronic Engineering and Computer Science at Queen Mary University of London. He is known for his work in software metrics and is the author of the textbook Software Metrics: A Rigorous Approach, as of 2014 in its third edition. == Education == Fenton received his bachelor's degree in mathematics from the London School of Economics in 1978. He earned his Master of Science in 1978 and Doctor of Philosophy in 1981 at the University of Sheffield. At Sheffield he was the second research student of Peter Vámos. His doctoral thesis was "Representations of Matroids". == Career == Fenton was a postdoctoral fellow in the mathematics department at University College Dublin from 1981 to 1982 and the Mathematics Institute of the University of Oxford from 1982 to 1984. At the end of that period he changed fields and began publishing papers on structured programming with Robin W. Whitty and Agnes A. Kaposi. In 1984, he joined the department of Electrical and Electronic Engineering at South Bank Polytechnic in London where he headed the Centre for Software and Systems Engineering research group. He began to publish on software metrics as well as program structure. In 1989, Fenton moved to City University as a reader in software reliability, and became a professor of Computing Science in 1992. In 1998, Fenton, along with Martin Neil and Ed Tranham, set up the company Agena Ltd in Cambridge. Fenton was CEO between 1998 and 2015 and remains a director. In 2000, Fenton joined Queen Mary University of London (School of Electronic Engineering and Computer Science) where he works part–time as a professor. He is director of the Risk and Information Management Research Group. == Selected publications == Textbooks Fenton, Norman E. (1991). Software Metrics: A Rigorous Approach. Chapman & Hall. ISBN 978-0-412-40440-5. OCLC 1069086327. Fenton, Norman E.; Pfleeger, Shari Lawrence (1997). Software Metrics: A Rigorous and Practical Approach (2 ed.). International Thomson Computer Press. ISBN 978-0-534-95600-4. OCLC 1055171425. Fenton, Norman; Bieman, James (1 October 2014). Software Metrics: A Rigorous and Practical Approach (3 ed.). Taylor & Francis. ISBN 978-1-4398-3822-8. OCLC 491888703. Fenton, Norman; Neil, Martin (3 September 2018). Risk Assessment and Decision Analysis with Bayesian Networks (2 ed.). CRC Press, Taylor & Francis Group. ISBN 978-1-138-03511-9. OCLC 1031043793. Articles Fenton, N. (1994). "Software measurement: a necessary scientific basis". IEEE Transactions on Software Engineering. 20 (3): 199–206. doi:10.1109/32.268921. Fenton, N.; Pfleeger, S.L.; Glass, R.L. (1994). "Science and substance: a challenge to software engineers". IEEE Software. 11 (4): 86–95. doi:10.1109/52.300094. S2CID 8528640. Kitchenham, B.; Pfleeger, S.L.; Fenton, N. (1995). "Towards a framework for software measurement validation". IEEE Transactions on Software Engineering. 21 (12): 929–944. doi:10.1109/32.489070. Fenton, N.E.; Neil, M. (1999). "A critique of software defect prediction models". IEEE Transactions on Software Engineering. 25 (5): 675–689. doi:10.1109/32.815326. Fenton, Norman E; Neil, Martin (1999). "Software metrics: successes, failures and new directions". Journal of Systems and Software. 47 (2–3): 149–157. doi:10.1016/S0164-1212(99)00035-7. Fenton, N.E.; Ohlsson, N. (2000). "Quantitative analysis of faults and failures in a complex software system". IEEE Transactions on Software Engineering. 26 (8): 797–814. doi:10.1109/32.879815. Fenton, Norman E.; Neil, Martin (2000). Software metrics: Roadmap. ICSE '00: Proceedings of the Conference on The Future of Software Engineering May 2000. pp. 357–370. doi:10.1145/336512.336588. Neil, Martin; Fenton, Norman; Nielson, Lars (2000). "Building large-scale Bayesian networks". The Knowledge Engineering Review. 15 (3): 257–284. doi:10.1017/S0269888900003039. S2CID 11309502. Fenton, Norman; Neil, Martin; Marsh, William; Hearty, Peter; Marquez, David; Krause, Paul; Mishra, Rajat (2007). "Predicting software defects in varying development lifecycles using Bayesian nets". Information and Software Technology. 49 (1): 32–43. doi:10.1016/j.infsof.2006.09.001. == References == == External links == Faculty website Norman Fenton publications indexed by Google Scholar
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Wikipedia:Norman H. Anning#0
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Norman Herbert Anning ((1883-08-28)August 28, 1883 – (1963-05-01)May 1, 1963) was a mathematician, assistant professor, professor emeritus, and instructor in mathematics, recognized and acclaimed in mathematics for publishing a proof of the characterization of the infinite sets of points in the plane with mutually integer distances, known as the Erdős–Anning theorem. == Life == Anning was originally from Holland Township (currently Chatsworth), Grey County, Ontario, Canada. In 1902, he won a scholarship to Queen's University, and received the Arts bachelor's degree in 1905, and the Arts master's degree in 1906 from the same institution. == Academic career == Anning served in the faculty of the University of Michigan since 1920, until he retired on 1953. From 1909 to 1910, he held a teaching position in the department of Mathematics and Science at Chilliwack High School, British Columbia. Anning was appointed as chairperson at the University of Michigan from 1951 to 1952, and treasurer secretary from 1925 to 1926 at the same institution. With Paul Erdős, he published a paper in 1945 containing what is now known as the Erdős–Anning theorem. The theorem states that an infinite number of points in the plane can have mutual integer distances only if all the points lie on a straight line. Anning retired on August 28, 1953. He died in Sunnydale, California on May 1, 1963. == Publications == Anning, N.H.; Erdős, P. (1945). "Integral distances". Bull. Amer. Math. Soc. 51 (8): 598–600. doi:10.1090/s0002-9904-1945-08407-9. Erdős, P.; Ruderman, HD; Willey, M.; Anning, N. (1935). "Problems for Solution: 3739-3743". The American Mathematical Monthly. 42 (6). JSTOR: 396–397. doi:10.2307/2301373. JSTOR 2301373. Norman H. Anning (1923). "Socrates Teaches Mathematics". School Science and Mathematics. 23 (6). Wiley Online Library: 581–584. doi:10.1111/j.1949-8594.1923.tb07353.x. Norman H. Anning (1917). "Another Method Of Deriving Sin 2α, sin 3α, And So On". School Science and Mathematics. 17 (1): 43–44. doi:10.1111/j.1949-8594.1917.tb01843.x. Norman H. Anning (1916). "Note On Triangles Whose Sides Are Whole Numbers". School Science and Mathematics. 16 (1): 82–83. doi:10.1111/j.1949-8594.1916.tb01570.x. Norman H. Anning (1915). "To Find Approximate Square Roots". School Science and Mathematics. 15 (3): 245–246. doi:10.1111/j.1949-8594.1915.tb10261.x. Norman H. Anning (1929). "What Are The Chances That; A Few Questions". School Science and Mathematics. 29 (5): 460. doi:10.1111/j.1949-8594.1929.tb02431.x. Norman H. Anning (1925). "A Device For Teachers Of Trigonometry". School Science and Mathematics. 25 (7): 739–740. doi:10.1111/j.1949-8594.1925.tb05056.x. == References ==
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Wikipedia:Norman L. Biggs#0
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Norman Linstead Biggs (born 2 January 1941) is a leading British mathematician focusing on discrete mathematics and in particular algebraic combinatorics. == Education == Biggs was educated at Harrow County Grammar School and then studied mathematics at Selwyn College, Cambridge. In 1962, Biggs gained first-class honours in his third year of the university's undergraduate degree in mathematics. 1946–1952: Uxendon Manor Primary School, Kenton, Middlesex 1952–1959: Harrow County Grammar School 1959–1963: Selwyn College, Cambridge (Entrance Exhibition 1959, Scholarship 1961) 1960: First Class, Mathematical Tripos Pt. I 1962: Wrangler, Mathematical Tripos Pt. II; B.A. (Cantab.) 1963: Distinction, Mathematical Tripos Pt. III 1988: D.Sc. (London); M.A. (Cantab.) == Career == He was a lecturer at University of Southampton, lecturer then reader at Royal Holloway, University of London, and Professor of Mathematics at the London School of Economics. He has been on the editorial board of a number of journals, including the Journal of Algebraic Combinatorics. He has been a member of the Council of the London Mathematical Society. He has written 12 books and over 100 papers on mathematical topics, many of them in algebraic combinatorics and its applications. He became Emeritus Professor in 2006 and continues to teach History of Mathematics in Finance and Economics for undergraduates. He is also vice-president of the British Society for the History of Mathematics. == Family == Biggs married Christine Mary Farmer in 1975 and has one daughter Clare Juliet born in 1980. == Interests and Hobbies == Biggs' interests include computational learning theory, the history of mathematics and historical metrology. Since 2006, he has been an emeritus professor at the London School of Economics. Biggs hobbies consist of writing about the history of weights and scales. He currently holds the position of Chair of the International Society of Antique Scale Collectors (Europe), and a member of the British Numismatic Society. == Work == === Mathematics === In 2002, Biggs wrote the second edition of Discrete Mathematics breaking down a wide range of topics into a clear and organised style. Biggs organised the book into four major sections; The Language of Mathematics, Techniques, Algorithms and Graphs, and Algebraic Methods. This book was an accumulation of Discrete Mathematics, first edition, textbook published in 1985 which dealt with calculations involving a finite number of steps rather than limiting processes. The second edition added nine new introductory chapters; Fundamental language of mathematicians, statements and proofs, the logical framework, sets and functions, and number system. This book stresses the significance of simple logical reasoning, shown by the exercises and examples given in the book. Each chapter contains modelled solutions, examples, exercises including hints and answers. === Algebraic Graph Theory === In 1974, Biggs published Algebraic Graph Theory which articulates properties of graphs in algebraic terms, then works out theorems regarding them. In the first section, he tackles the applications of linear algebra and matrix theory; algebraic constructions such as adjacency matrix and the incidence matrix and their applications are discussed in depth. Next, there is a wide-ranging description of the theory of chromatic polynomials. The last section discusses symmetry and regularity properties. Biggs makes important connections with other branches of algebraic combinatorics and group theory. === Computational Learning Theory === In 1997, N. Biggs and M. Anthony wrote a book titled Computational Learning Theory: an Introduction. Both Biggs and Anthony focused on the necessary background material from logic, probability, and complex theory. This book is an introduction to computational learning. === History of Mathematics === Biggs contributed to thirteen journals and books developing topics such as the four-colour conjecture, the roots/history of combinatorics, calculus, Topology on the 19th century, and mathematicians. In addition, Biggs examined the ideas of William Ludlam, Thomas Harriot, John Arbuthnot, and Leonhard Euler. == Chip-Firing Game == The chip-firing game has been around for less than 20 years. It has become an important part of the study of structural combinatorics. The set of configurations that are stable and recurrent for this game can be given the structure of an abelian group. In addition, the order of the group is equal to the tree number of the graph. == Publications == === Summary of Biggs' published Books on Mathematics === Finite Groups of Automorphisms, Cambridge University Press (1971) Algebraic Graph Theory, Cambridge University Press (1974) Graph Theory, 1736–1936 (with E.K. Lloyd and R.J. Wilson), Oxford University Press (1976) (Japanese edition 1986) Interaction Models, Cambridge University Press (1977) Permutation Groups and Combinatorial Structures (with A.T. White), Cambridge University Press, (1979), (Chinese edition 1988) Discrete Mathematics, Oxford University Press (1989) (Spanish edition 1994) Introduction to Computing with Pascal, Oxford University Press (1989) Computational Learning Theory: an Introduction (with M. Anthony) (1997) Algebraic Graph Theory (Second Edition), Cambridge University Press (1993) Mathematics for Economics and Finance (with M. Anthony), Cambridge University Press (1996) (Chinese edition 1998; Japanese edition 2000) Discrete Mathematics, (Second Edition), Oxford University Press (2002) Codes: An Introduction to Information Communication and Cryptography, Springer Verlag (2008) === Summary of Biggs' latest published Papers on Mathematics === 2000 'A matrix method for chromatic polynomials – II', CDAM Research Report Series, LSE-CDAM 2000–04, April 2000. (with P.Reinfeld), 'The chromatic roots of generalised dodecahedra', CDAM Research Report Series, LSE-CDAM 2000–07, June 2000. 2001 'Equimodular curves for reducible matrices', CDAM Research Report Series, LSE-CDAM 2001–01, January 2001. 'A matrix method for chromatic polynomials', Journal of Combinatorial Theory, Series B, 82 (2001) 19–29. 2002 'Chromatic polynomials for twisted bracelets', Bull. London Math. Soc. 34 (2002) 129–139. 'Chromatic polynomials and representations of the symmetric group', Linear Algebra and its Applications 356 (2002) 3–26. 'Equimodular curves', Discrete Mathematics 259 (2002) 37–57. 2004 'Algebraic methods for chromatic polynomials' (with M H Klin and P Reinfeld), Europ. J. Combinatorics 25 (2004) 147–160. 'Specht modules and chromatic polynomials', Journal of Combinatorial Theory, Series B 92 (2004) 359 – 377. 2005 'Chromatic polynomials of some families of graphs I: Theorems and Conjectures', CDAM Research Report Series, LSE-CDAM 2005–09, May 2005. 2007 'The critical group from a cryptographic perspective', Bull. London Math. Soc., 39 (2007) 829–836. 2008 'Chromatic Roots of the Quartic Mobius Ladders', CDAM Research Report LSE-CDAM 2008–05, May 2008. 'A Matrix Method for Flow Polynomials', CDAM Research Report LSE-CDAM 2008–08, June 2008. 2009 'Tutte Polynomials of Bracelets', CDAM Research Report LSE-CDAM-2009-01, January 2009. 'Strongly Regular Graphs with No Triangles', Research Report, September 2009. arXiv:0911.2160v1 'Families of Parameters for SRNT Graphs', Research Report, October 2009. arXiv:0911.2455v1 2010 'Tutte Polynomials of Bracelets', J. Algebraic Combinatorics 32 (2010) 389–398. 'The Second Subconstituent of some Strongly Regular Graphs', Research Report, February 2010. arXiv:1003.0175v1 2011 'Some Properties of Strongly Regular Graphs', Research Report, May 2011. arXiv:1106.0889v1 For other published work on the history of mathematics, please see. == See also == Computational learning theory Four color theorem == References == == External links == Norman Biggs personal web page at LSE Norman Linstead Biggs at the Mathematics Genealogy Project Cambridge University Press: Norman L Biggs
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Wikipedia:Normed algebra#0
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In mathematics, a normed algebra A is an algebra over a field which has a sub-multiplicative norm: ∀ x , y ∈ A ‖ x y ‖ ≤ ‖ x ‖ ‖ y ‖ . {\displaystyle \forall x,y\in A\qquad \|xy\|\leq \|x\|\|y\|.} Some authors require it to have a multiplicative identity 1A such that ║1A║ = 1. == See also == == External reading == "Normed Algebra". Encyclopaedia of Mathematics. Retrieved 20 May 2018.
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Wikipedia:Nova fractal#0
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The Newton fractal is a boundary set in the complex plane which is characterized by Newton's method applied to a fixed polynomial p(z) ∈ C {\displaystyle \mathbb {C} } [z] or transcendental function. It is the Julia set of the meromorphic function z ↦ z − p(z)/p′(z) which is given by Newton's method. When there are no attractive cycles (of order greater than 1), it divides the complex plane into regions Gk, each of which is associated with a root ζk of the polynomial, k = 1, …, deg(p). In this way the Newton fractal is similar to the Mandelbrot set, and like other fractals it exhibits an intricate appearance arising from a simple description. It is relevant to numerical analysis because it shows that (outside the region of quadratic convergence) the Newton method can be very sensitive to its choice of start point. Almost all points of the complex plane are associated with one of the deg(p) roots of a given polynomial in the following way: the point is used as starting value z0 for Newton's iteration zn + 1 := zn − p(zn)/p'(zn), yielding a sequence of points z1, z2, …, If the sequence converges to the root ζk, then z0 was an element of the region Gk. However, for every polynomial of degree at least 2 there are points for which the Newton iteration does not converge to any root: examples are the boundaries of the basins of attraction of the various roots. There are even polynomials for which open sets of starting points fail to converge to any root: a simple example is z3 − 2z + 2, where some points are attracted by the cycle 0, 1, 0, 1… rather than by a root. An open set for which the iterations converge towards a given root or cycle (that is not a fixed point), is a Fatou set for the iteration. The complementary set to the union of all these, is the Julia set. The Fatou sets have common boundary, namely the Julia set. Therefore, each point of the Julia set is a point of accumulation for each of the Fatou sets. It is this property that causes the fractal structure of the Julia set (when the degree of the polynomial is larger than 2). To plot images of the fractal, one may first choose a specified number d of complex points (ζ1, …, ζd) and compute the coefficients (p1, …, pd) of the polynomial p ( z ) = z d + p 1 z d − 1 + ⋯ + p d − 1 z + p d := ( z − ζ 1 ) ( z − ζ 2 ) ⋯ ( z − ζ d ) {\displaystyle p(z)=z^{d}+p_{1}z^{d-1}+\cdots +p_{d-1}z+p_{d}:=(z-\zeta _{1})(z-\zeta _{2})\cdots (z-\zeta _{d})} . Then for a rectangular lattice z m n = z 00 + m Δ x + i n Δ y ; m = 0 , … , M − 1 ; n = 0 , … , N − 1 {\displaystyle z_{mn}=z_{00}+m\,\Delta x+in\,\Delta y;\quad m=0,\ldots ,M-1;\quad n=0,\ldots ,N-1} of points in C {\displaystyle \mathbb {C} } , one finds the index k(m,n) of the corresponding root ζk(m,n) and uses this to fill an M × N raster grid by assigning to each point (m,n) a color fk(m,n). Additionally or alternatively the colors may be dependent on the distance D(m,n), which is defined to be the first value D such that |zD − ζk(m,n)| < ε for some previously fixed small ε > 0. == Generalization of Newton fractals == A generalization of Newton's iteration is z n + 1 = z n − a p ( z n ) p ′ ( z n ) {\displaystyle z_{n+1}=z_{n}-a{\frac {p(z_{n})}{p'(z_{n})}}} where a is any complex number. The special choice a = 1 corresponds to the Newton fractal. The fixed points of this map are stable when a lies inside the disk of radius 1 centered at 1. When a is outside this disk, the fixed points are locally unstable, however the map still exhibits a fractal structure in the sense of Julia set. If p is a polynomial of degree d, then the sequence zn is bounded provided that a is inside a disk of radius d centered at d. More generally, Newton's fractal is a special case of a Julia set. Serie : p(z) = zn- 1 Other fractals where potential and trigonometric functions are multiplied. p(z) = zn*Sin(z) - 1 === Nova fractal === The Nova fractal invented in the mid 1990s by Paul Derbyshire, is a generalization of the Newton fractal with the addition of a value c at each step: z n + 1 = z n − a p ( z n ) p ′ ( z n ) + c = G ( a , c , z ) {\displaystyle z_{n+1}=z_{n}-a{\frac {p(z_{n})}{p'(z_{n})}}+c=G(a,c,z)} The "Julia" variant of the Nova fractal keeps c constant over the image and initializes z0 to the pixel coordinates. The "Mandelbrot" variant of the Nova fractal initializes c to the pixel coordinates and sets z0 to a critical point, where ∂ ∂ z G ( a , c , z ) = 0. {\displaystyle {\frac {\partial }{\partial z}}G(a,c,z)=0.} Commonly-used polynomials like p(z) = z3 − 1 or p(z) = (z − 1)3 lead to a critical point at z = 1. == Implementation == In order to implement the Newton fractal, it is necessary to have a starting function as well as its derivative function: f ( z ) = z 3 − 1 f ′ ( z ) = 3 z 2 {\displaystyle {\begin{aligned}f(z)&=z^{3}-1\\f'(z)&=3z^{2}\end{aligned}}} The three roots of the function are z = 1 , − 1 2 + 3 2 i , − 1 2 − 3 2 i {\displaystyle z=1,\ -{\tfrac {1}{2}}+{\tfrac {\sqrt {3}}{2}}i,\ -{\tfrac {1}{2}}-{\tfrac {\sqrt {3}}{2}}i} The above-defined functions can be translated in pseudocode as follows: It is now just a matter of implementing the Newton method using the given functions. == See also == Julia set Mandelbrot set == References == == Further reading == J. H. Hubbard, D. Schleicher, S. Sutherland: How to Find All Roots of Complex Polynomials by Newton's Method, Inventiones Mathematicae vol. 146 (2001) – with a discussion of the global structure of Newton fractals On the Number of Iterations for Newton's Method by Dierk Schleicher July 21, 2000 Newton's Method as a Dynamical System by Johannes Rueckert Newton's Fractal (which Newton knew nothing about) by 3Blue1Brown, along with an interactive demonstration of the fractal on his website, and the source code for the demonstration
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Wikipedia:Nowhere continuous function#0
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In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f {\displaystyle f} is a function from real numbers to real numbers, then f {\displaystyle f} is nowhere continuous if for each point x {\displaystyle x} there is some ε > 0 {\displaystyle \varepsilon >0} such that for every δ > 0 , {\displaystyle \delta >0,} we can find a point y {\displaystyle y} such that | x − y | < δ {\displaystyle |x-y|<\delta } and | f ( x ) − f ( y ) | ≥ ε {\displaystyle |f(x)-f(y)|\geq \varepsilon } . Therefore, no matter how close it gets to any fixed point, there are even closer points at which the function takes not-nearby values. More general definitions of this kind of function can be obtained, by replacing the absolute value by the distance function in a metric space, or by using the definition of continuity in a topological space. == Examples == === Dirichlet function === One example of such a function is the indicator function of the rational numbers, also known as the Dirichlet function. This function is denoted as 1 Q {\displaystyle \mathbf {1} _{\mathbb {Q} }} and has domain and codomain both equal to the real numbers. By definition, 1 Q ( x ) {\displaystyle \mathbf {1} _{\mathbb {Q} }(x)} is equal to 1 {\displaystyle 1} if x {\displaystyle x} is a rational number and it is 0 {\displaystyle 0} otherwise. More generally, if E {\displaystyle E} is any subset of a topological space X {\displaystyle X} such that both E {\displaystyle E} and the complement of E {\displaystyle E} are dense in X , {\displaystyle X,} then the real-valued function which takes the value 1 {\displaystyle 1} on E {\displaystyle E} and 0 {\displaystyle 0} on the complement of E {\displaystyle E} will be nowhere continuous. Functions of this type were originally investigated by Peter Gustav Lejeune Dirichlet. === Non-trivial additive functions === A function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is called an additive function if it satisfies Cauchy's functional equation: f ( x + y ) = f ( x ) + f ( y ) for all x , y ∈ R . {\displaystyle f(x+y)=f(x)+f(y)\quad {\text{ for all }}x,y\in \mathbb {R} .} For example, every map of form x ↦ c x , {\displaystyle x\mapsto cx,} where c ∈ R {\displaystyle c\in \mathbb {R} } is some constant, is additive (in fact, it is linear and continuous). Furthermore, every linear map L : R → R {\displaystyle L:\mathbb {R} \to \mathbb {R} } is of this form (by taking c := L ( 1 ) {\displaystyle c:=L(1)} ). Although every linear map is additive, not all additive maps are linear. An additive map f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is linear if and only if there exists a point at which it is continuous, in which case it is continuous everywhere. Consequently, every non-linear additive function R → R {\displaystyle \mathbb {R} \to \mathbb {R} } is discontinuous at every point of its domain. Nevertheless, the restriction of any additive function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } to any real scalar multiple of the rational numbers Q {\displaystyle \mathbb {Q} } is continuous; explicitly, this means that for every real r ∈ R , {\displaystyle r\in \mathbb {R} ,} the restriction f | r Q : r Q → R {\displaystyle f{\big \vert }_{r\mathbb {Q} }:r\,\mathbb {Q} \to \mathbb {R} } to the set r Q := { r q : q ∈ Q } {\displaystyle r\,\mathbb {Q} :=\{rq:q\in \mathbb {Q} \}} is a continuous function. Thus if f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is a non-linear additive function then for every point x ∈ R , {\displaystyle x\in \mathbb {R} ,} f {\displaystyle f} is discontinuous at x {\displaystyle x} but x {\displaystyle x} is also contained in some dense subset D ⊆ R {\displaystyle D\subseteq \mathbb {R} } on which f {\displaystyle f} 's restriction f | D : D → R {\displaystyle f\vert _{D}:D\to \mathbb {R} } is continuous (specifically, take D := x Q {\displaystyle D:=x\,\mathbb {Q} } if x ≠ 0 , {\displaystyle x\neq 0,} and take D := Q {\displaystyle D:=\mathbb {Q} } if x = 0 {\displaystyle x=0} ). === Discontinuous linear maps === A linear map between two topological vector spaces, such as normed spaces for example, is continuous (everywhere) if and only if there exists a point at which it is continuous, in which case it is even uniformly continuous. Consequently, every linear map is either continuous everywhere or else continuous nowhere. Every linear functional is a linear map and on every infinite-dimensional normed space, there exists some discontinuous linear functional. === Other functions === The Conway base 13 function is discontinuous at every point. == Hyperreal characterisation == A real function f {\displaystyle f} is nowhere continuous if its natural hyperreal extension has the property that every x {\displaystyle x} is infinitely close to a y {\displaystyle y} such that the difference f ( x ) − f ( y ) {\displaystyle f(x)-f(y)} is appreciable (that is, not infinitesimal). == See also == Blumberg theorem – even if a real function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is nowhere continuous, there is a dense subset D {\displaystyle D} of R {\displaystyle \mathbb {R} } such that the restriction of f {\displaystyle f} to D {\displaystyle D} is continuous. Thomae's function (also known as the popcorn function) – a function that is continuous at all irrational numbers and discontinuous at all rational numbers. Weierstrass function – a function continuous everywhere (inside its domain) and differentiable nowhere. == References == == External links == "Dirichlet-function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Dirichlet Function — from MathWorld The Modified Dirichlet Function Archived 2019-05-02 at the Wayback Machine by George Beck, The Wolfram Demonstrations Project.
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Wikipedia:Nth root#0
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In mathematics, an nth root of a number x is a number r which, when raised to the power of n, yields x: r n = r × r × ⋯ × r ⏟ n factors = x . {\displaystyle r^{n}=\underbrace {r\times r\times \dotsb \times r} _{n{\text{ factors}}}=x.} The positive integer n is called the index or degree, and the number x of which the root is taken is the radicand. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an nth root is a root extraction. For example, 3 is a square root of 9, since 32 = 9, and −3 is also a square root of 9, since (−3)2 = 9. The nth root of x is written as x n {\displaystyle {\sqrt[{n}]{x}}} using the radical symbol x {\displaystyle {\sqrt {\phantom {x}}}} . The square root is usually written as x {\displaystyle {\sqrt {x}}} , with the degree omitted. Taking the nth root of a number, for fixed n {\displaystyle n} , is the inverse of raising a number to the nth power, and can be written as a fractional exponent: x n = x 1 / n . {\displaystyle {\sqrt[{n}]{x}}=x^{1/n}.} For a positive real number x, x {\displaystyle {\sqrt {x}}} denotes the positive square root of x and x n {\displaystyle {\sqrt[{n}]{x}}} denotes the positive real nth root. A negative real number −x has no real-valued square roots, but when x is treated as a complex number it has two imaginary square roots, + i x {\displaystyle +i{\sqrt {x}}} and − i x {\displaystyle -i{\sqrt {x}}} , where i is the imaginary unit. In general, any non-zero complex number has n distinct complex-valued nth roots, equally distributed around a complex circle of constant absolute value. (The nth root of 0 is zero with multiplicity n, and this circle degenerates to a point.) Extracting the nth roots of a complex number x can thus be taken to be a multivalued function. By convention the principal value of this function, called the principal root and denoted x n {\displaystyle {\sqrt[{n}]{x}}} , is taken to be the nth root with the greatest real part and in the special case when x is a negative real number, the one with a positive imaginary part. The principal root of a positive real number is thus also a positive real number. As a function, the principal root is continuous in the whole complex plane, except along the negative real axis. An unresolved root, especially one using the radical symbol, is sometimes referred to as a surd or a radical. Any expression containing a radical, whether it is a square root, a cube root, or a higher root, is called a radical expression, and if it contains no transcendental functions or transcendental numbers it is called an algebraic expression. Roots are used for determining the radius of convergence of a power series with the root test. The nth roots of 1 are called roots of unity and play a fundamental role in various areas of mathematics, such as number theory, theory of equations, and Fourier transform. == History == An archaic term for the operation of taking nth roots is radication. == Definition and notation == An nth root of a number x, where n is a positive integer, is any of the n real or complex numbers r whose nth power is x: r n = x . {\displaystyle r^{n}=x.} Every positive real number x has a single positive nth root, called the principal nth root, which is written x n {\displaystyle {\sqrt[{n}]{x}}} . For n equal to 2 this is called the principal square root and the n is omitted. The nth root can also be represented using exponentiation as x1/n. For even values of n, positive numbers also have a negative nth root, while negative numbers do not have a real nth root. For odd values of n, every negative number x has a real negative nth root. For example, −2 has a real 5th root, − 2 5 = − 1.148698354 … {\displaystyle {\sqrt[{5}]{-2}}=-1.148698354\ldots } but −2 does not have any real 6th roots. Every non-zero number x, real or complex, has n different complex number nth roots. (In the case x is real, this count includes any real nth roots.) The only complex root of 0 is 0. The nth roots of almost all numbers (all integers except the nth powers, and all rationals except the quotients of two nth powers) are irrational. For example, 2 = 1.414213562 … {\displaystyle {\sqrt {2}}=1.414213562\ldots } All nth roots of rational numbers are algebraic numbers, and all nth roots of integers are algebraic integers. The term "surd" traces back to Al-Khwarizmi (c. 825), who referred to rational and irrational numbers as audible and inaudible, respectively. This later led to the Arabic word أصم (asamm, meaning "deaf" or "dumb") for irrational number being translated into Latin as surdus (meaning "deaf" or "mute"). Gerard of Cremona (c. 1150), Fibonacci (1202), and then Robert Recorde (1551) all used the term to refer to unresolved irrational roots, that is, expressions of the form r n {\displaystyle {\sqrt[{n}]{r}}} , in which n {\displaystyle n} and r {\displaystyle r} are integer numerals and the whole expression denotes an irrational number. Irrational numbers of the form ± a , {\displaystyle \pm {\sqrt {a}},} where a {\displaystyle a} is rational, are called pure quadratic surds; irrational numbers of the form a ± b {\displaystyle a\pm {\sqrt {b}}} , where a {\displaystyle a} and b {\displaystyle b} are rational, are called mixed quadratic surds. === Square roots === A square root of a number x is a number r which, when squared, becomes x: r 2 = x . {\displaystyle r^{2}=x.} Every positive real number has two square roots, one positive and one negative. For example, the two square roots of 25 are 5 and −5. The positive square root is also known as the principal square root, and is denoted with a radical sign: 25 = 5. {\displaystyle {\sqrt {25}}=5.} Since the square of every real number is nonnegative, negative numbers do not have real square roots. However, for every negative real number there are two imaginary square roots. For example, the square roots of −25 are 5i and −5i, where i represents a number whose square is −1. === Cube roots === A cube root of a number x is a number r whose cube is x: r 3 = x . {\displaystyle r^{3}=x.} Every real number x has exactly one real cube root, written x 3 {\displaystyle {\sqrt[{3}]{x}}} . For example, 8 3 = 2 − 8 3 = − 2. {\displaystyle {\begin{aligned}{\sqrt[{3}]{8}}&=2\\{\sqrt[{3}]{-8}}&=-2.\end{aligned}}} Every real number has two additional complex cube roots. == Identities and properties == Expressing the degree of an nth root in its exponent form, as in x 1 / n {\displaystyle x^{1/n}} , makes it easier to manipulate powers and roots. If a {\displaystyle a} is a non-negative real number, a m n = ( a m ) 1 / n = a m / n = ( a 1 / n ) m = ( a n ) m . {\displaystyle {\sqrt[{n}]{a^{m}}}=(a^{m})^{1/n}=a^{m/n}=(a^{1/n})^{m}=({\sqrt[{n}]{a}})^{m}.} Every non-negative number has exactly one non-negative real nth root, and so the rules for operations with surds involving non-negative radicands a {\displaystyle a} and b {\displaystyle b} are straightforward within the real numbers: a b n = a n b n a b n = a n b n {\displaystyle {\begin{aligned}{\sqrt[{n}]{ab}}&={\sqrt[{n}]{a}}{\sqrt[{n}]{b}}\\{\sqrt[{n}]{\frac {a}{b}}}&={\frac {\sqrt[{n}]{a}}{\sqrt[{n}]{b}}}\end{aligned}}} Subtleties can occur when taking the nth roots of negative or complex numbers. For instance: − 1 × − 1 ≠ − 1 × − 1 = 1 , {\displaystyle {\sqrt {-1}}\times {\sqrt {-1}}\neq {\sqrt {-1\times -1}}=1,\quad } but, rather, − 1 × − 1 = i × i = i 2 = − 1. {\displaystyle \quad {\sqrt {-1}}\times {\sqrt {-1}}=i\times i=i^{2}=-1.} Since the rule a n × b n = a b n {\displaystyle {\sqrt[{n}]{a}}\times {\sqrt[{n}]{b}}={\sqrt[{n}]{ab}}} strictly holds for non-negative real radicands only, its application leads to the inequality in the first step above. == Simplified form of a radical expression == A non-nested radical expression is said to be in simplified form if no factor of the radicand can be written as a power greater than or equal to the index; there are no fractions inside the radical sign; and there are no radicals in the denominator. For example, to write the radical expression 32 / 5 {\displaystyle \textstyle {\sqrt {32/5}}} in simplified form, we can proceed as follows. First, look for a perfect square under the square root sign and remove it: 32 5 = 16 ⋅ 2 5 = 16 ⋅ 2 5 = 4 2 5 {\displaystyle {\sqrt {\frac {32}{5}}}={\sqrt {\frac {16\cdot 2}{5}}}={\sqrt {16}}\cdot {\sqrt {\frac {2}{5}}}=4{\sqrt {\frac {2}{5}}}} Next, there is a fraction under the radical sign, which we change as follows: 4 2 5 = 4 2 5 {\displaystyle 4{\sqrt {\frac {2}{5}}}={\frac {4{\sqrt {2}}}{\sqrt {5}}}} Finally, we remove the radical from the denominator as follows: 4 2 5 = 4 2 5 ⋅ 5 5 = 4 10 5 = 4 5 10 {\displaystyle {\frac {4{\sqrt {2}}}{\sqrt {5}}}={\frac {4{\sqrt {2}}}{\sqrt {5}}}\cdot {\frac {\sqrt {5}}{\sqrt {5}}}={\frac {4{\sqrt {10}}}{5}}={\frac {4}{5}}{\sqrt {10}}} When there is a denominator involving surds it is always possible to find a factor to multiply both numerator and denominator by to simplify the expression. For instance using the factorization of the sum of two cubes: 1 a 3 + b 3 = a 2 3 − a b 3 + b 2 3 ( a 3 + b 3 ) ( a 2 3 − a b 3 + b 2 3 ) = a 2 3 − a b 3 + b 2 3 a + b . {\displaystyle {\frac {1}{{\sqrt[{3}]{a}}+{\sqrt[{3}]{b}}}}={\frac {{\sqrt[{3}]{a^{2}}}-{\sqrt[{3}]{ab}}+{\sqrt[{3}]{b^{2}}}}{\left({\sqrt[{3}]{a}}+{\sqrt[{3}]{b}}\right)\left({\sqrt[{3}]{a^{2}}}-{\sqrt[{3}]{ab}}+{\sqrt[{3}]{b^{2}}}\right)}}={\frac {{\sqrt[{3}]{a^{2}}}-{\sqrt[{3}]{ab}}+{\sqrt[{3}]{b^{2}}}}{a+b}}.} Simplifying radical expressions involving nested radicals can be quite difficult. In particular, denesting is not always possible, and when possible, it may involve advanced Galois theory. Moreover, when complete denesting is impossible, there is no general canonical form such that the equality of two numbers can be tested by simply looking at their canonical expressions. For example, it is not obvious that 3 + 2 2 = 1 + 2 . {\displaystyle {\sqrt {3+2{\sqrt {2}}}}=1+{\sqrt {2}}.} The above can be derived through: 3 + 2 2 = 1 + 2 2 + 2 = 1 2 + 2 2 + 2 2 = ( 1 + 2 ) 2 = 1 + 2 {\displaystyle {\sqrt {3+2{\sqrt {2}}}}={\sqrt {1+2{\sqrt {2}}+2}}={\sqrt {1^{2}+2{\sqrt {2}}+{\sqrt {2}}^{2}}}={\sqrt {\left(1+{\sqrt {2}}\right)^{2}}}=1+{\sqrt {2}}} Let r = p / q {\displaystyle r=p/q} , with p and q coprime and positive integers. Then r n = p n / q n {\displaystyle {\sqrt[{n}]{r}}={\sqrt[{n}]{p}}/{\sqrt[{n}]{q}}} is rational if and only if both p n {\displaystyle {\sqrt[{n}]{p}}} and q n {\displaystyle {\sqrt[{n}]{q}}} are integers, which means that both p and q are nth powers of some integer. == Infinite series == The radical or root may be represented by the infinite series: ( 1 + x ) s t = ∑ n = 0 ∞ ∏ k = 0 n − 1 ( s − k t ) n ! t n x n {\displaystyle (1+x)^{\frac {s}{t}}=\sum _{n=0}^{\infty }{\frac {\prod _{k=0}^{n-1}(s-kt)}{n!t^{n}}}x^{n}} with | x | < 1 {\displaystyle |x|<1} . This expression can be derived from the binomial series. == Computing principal roots == === Using Newton's method === The nth root of a number A can be computed with Newton's method, which starts with an initial guess x0 and then iterates using the recurrence relation x k + 1 = x k − x k n − A n x k n − 1 {\displaystyle x_{k+1}=x_{k}-{\frac {x_{k}^{n}-A}{nx_{k}^{n-1}}}} until the desired precision is reached. For computational efficiency, the recurrence relation is commonly rewritten x k + 1 = n − 1 n x k + A n 1 x k n − 1 . {\displaystyle x_{k+1}={\frac {n-1}{n}}\,x_{k}+{\frac {A}{n}}\,{\frac {1}{x_{k}^{n-1}}}.} This allows to have only one exponentiation, and to compute once for all the first factor of each term. For example, to find the fifth root of 34, we plug in n = 5, A = 34 and x0 = 2 (initial guess). The first 5 iterations are, approximately: (All correct digits shown.) The approximation x4 is accurate to 25 decimal places and x5 is good for 51. Newton's method can be modified to produce various generalized continued fractions for the nth root. For example, z n = x n + y n = x + y n x n − 1 + ( n − 1 ) y 2 x + ( n + 1 ) y 3 n x n − 1 + ( 2 n − 1 ) y 2 x + ( 2 n + 1 ) y 5 n x n − 1 + ( 3 n − 1 ) y 2 x + ⋱ . {\displaystyle {\sqrt[{n}]{z}}={\sqrt[{n}]{x^{n}+y}}=x+{\cfrac {y}{nx^{n-1}+{\cfrac {(n-1)y}{2x+{\cfrac {(n+1)y}{3nx^{n-1}+{\cfrac {(2n-1)y}{2x+{\cfrac {(2n+1)y}{5nx^{n-1}+{\cfrac {(3n-1)y}{2x+\ddots }}}}}}}}}}}}.} === Digit-by-digit calculation of principal roots of decimal (base 10) numbers === Building on the digit-by-digit calculation of a square root, it can be seen that the formula used there, x ( 20 p + x ) ≤ c {\displaystyle x(20p+x)\leq c} , or x 2 + 20 x p ≤ c {\displaystyle x^{2}+20xp\leq c} , follows a pattern involving Pascal's triangle. For the nth root of a number P ( n , i ) {\displaystyle P(n,i)} is defined as the value of element i {\displaystyle i} in row n {\displaystyle n} of Pascal's Triangle such that P ( 4 , 1 ) = 4 {\displaystyle P(4,1)=4} , we can rewrite the expression as ∑ i = 0 n − 1 10 i P ( n , i ) p i x n − i {\displaystyle \sum _{i=0}^{n-1}10^{i}P(n,i)p^{i}x^{n-i}} . For convenience, call the result of this expression y {\displaystyle y} . Using this more general expression, any positive principal root can be computed, digit-by-digit, as follows. Write the original number in decimal form. The numbers are written similar to the long division algorithm, and, as in long division, the root will be written on the line above. Now separate the digits into groups of digits equating to the root being taken, starting from the decimal point and going both left and right. The decimal point of the root will be above the decimal point of the radicand. One digit of the root will appear above each group of digits of the original number. Beginning with the left-most group of digits, do the following procedure for each group: Starting on the left, bring down the most significant (leftmost) group of digits not yet used (if all the digits have been used, write "0" the number of times required to make a group) and write them to the right of the remainder from the previous step (on the first step, there will be no remainder). In other words, multiply the remainder by 10 n {\displaystyle 10^{n}} and add the digits from the next group. This will be the current value c. Find p and x, as follows: Let p {\displaystyle p} be the part of the root found so far, ignoring any decimal point. (For the first step, p = 0 {\displaystyle p=0} and 0 0 = 1 {\displaystyle 0^{0}=1} ). Determine the greatest digit x {\displaystyle x} such that y ≤ c {\displaystyle y\leq c} . Place the digit x {\displaystyle x} as the next digit of the root, i.e., above the group of digits you just brought down. Thus the next p will be the old p times 10 plus x. Subtract y {\displaystyle y} from c {\displaystyle c} to form a new remainder. If the remainder is zero and there are no more digits to bring down, then the algorithm has terminated. Otherwise go back to step 1 for another iteration. ==== Examples ==== Find the square root of 152.2756. 1 2. 3 4 / \/ 01 52.27 56 (Results) (Explanations) 01 x = 1 100·1·00·12 + 101·2·01·11 ≤ 1 < 100·1·00·22 + 101·2·01·21 01 y = 1 y = 100·1·00·12 + 101·2·01·11 = 1 + 0 = 1 00 52 x = 2 100·1·10·22 + 101·2·11·21 ≤ 52 < 100·1·10·32 + 101·2·11·31 00 44 y = 44 y = 100·1·10·22 + 101·2·11·21 = 4 + 40 = 44 08 27 x = 3 100·1·120·32 + 101·2·121·31 ≤ 827 < 100·1·120·42 + 101·2·121·41 07 29 y = 729 y = 100·1·120·32 + 101·2·121·31 = 9 + 720 = 729 98 56 x = 4 100·1·1230·42 + 101·2·1231·41 ≤ 9856 < 100·1·1230·52 + 101·2·1231·51 98 56 y = 9856 y = 100·1·1230·42 + 101·2·1231·41 = 16 + 9840 = 9856 00 00 Algorithm terminates: Answer is 12.34 Find the cube root of 4192 truncated to the nearest thousandth. 1 6. 1 2 4 3 / \/ 004 192.000 000 000 (Results) (Explanations) 004 x = 1 100·1·00·13 + 101·3·01·12 + 102·3·02·11 ≤ 4 < 100·1·00·23 + 101·3·01·22 + 102·3·02·21 001 y = 1 y = 100·1·00·13 + 101·3·01·12 + 102·3·02·11 = 1 + 0 + 0 = 1 003 192 x = 6 100·1·10·63 + 101·3·11·62 + 102·3·12·61 ≤ 3192 < 100·1·10·73 + 101·3·11·72 + 102·3·12·71 003 096 y = 3096 y = 100·1·10·63 + 101·3·11·62 + 102·3·12·61 = 216 + 1,080 + 1,800 = 3,096 096 000 x = 1 100·1·160·13 + 101·3·161·12 + 102·3·162·11 ≤ 96000 < 100·1·160·23 + 101·3·161·22 + 102·3·162·21 077 281 y = 77281 y = 100·1·160·13 + 101·3·161·12 + 102·3·162·11 = 1 + 480 + 76,800 = 77,281 018 719 000 x = 2 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 ≤ 18719000 < 100·1·1610·33 + 101·3·1611·32 + 102·3·1612·31 015 571 928 y = 15571928 y = 100·1·1610·23 + 101·3·1611·22 + 102·3·1612·21 = 8 + 19,320 + 15,552,600 = 15,571,928 003 147 072 000 x = 4 100·1·16120·43 + 101·3·16121·42 + 102·3·16122·41 ≤ 3147072000 < 100·1·16120·53 + 101·3·16121·52 + 102·3·16122·51 The desired precision is achieved. The cube root of 4192 is 16.124... === Logarithmic calculation === The principal nth root of a positive number can be computed using logarithms. Starting from the equation that defines r as an nth root of x, namely r n = x , {\displaystyle r^{n}=x,} with x positive and therefore its principal root r also positive, one takes logarithms of both sides (any base of the logarithm will do) to obtain n log b r = log b x hence log b r = log b x n . {\displaystyle n\log _{b}r=\log _{b}x\quad \quad {\text{hence}}\quad \quad \log _{b}r={\frac {\log _{b}x}{n}}.} The root r is recovered from this by taking the antilog: r = b 1 n log b x . {\displaystyle r=b^{{\frac {1}{n}}\log _{b}x}.} (Note: That formula shows b raised to the power of the result of the division, not b multiplied by the result of the division.) For the case in which x is negative and n is odd, there is one real root r which is also negative. This can be found by first multiplying both sides of the defining equation by −1 to obtain | r | n = | x | , {\displaystyle |r|^{n}=|x|,} then proceeding as before to find |r|, and using r = −|r|. == Geometric constructibility == The ancient Greek mathematicians knew how to use compass and straightedge to construct a length equal to the square root of a given length, when an auxiliary line of unit length is given. In 1837 Pierre Wantzel proved that an nth root of a given length cannot be constructed if n is not a power of 2. == Complex roots == Every complex number other than 0 has n different nth roots. === Square roots === The two square roots of a complex number are always negatives of each other. For example, the square roots of −4 are 2i and −2i, and the square roots of i are 1 2 ( 1 + i ) and − 1 2 ( 1 + i ) . {\displaystyle {\tfrac {1}{\sqrt {2}}}(1+i)\quad {\text{and}}\quad -{\tfrac {1}{\sqrt {2}}}(1+i).} If we express a complex number in polar form, then the square root can be obtained by taking the square root of the radius and halving the angle: r e i θ = ± r ⋅ e i θ / 2 . {\displaystyle {\sqrt {re^{i\theta }}}=\pm {\sqrt {r}}\cdot e^{i\theta /2}.} A principal root of a complex number may be chosen in various ways, for example r e i θ = r ⋅ e i θ / 2 {\displaystyle {\sqrt {re^{i\theta }}}={\sqrt {r}}\cdot e^{i\theta /2}} which introduces a branch cut in the complex plane along the positive real axis with the condition 0 ≤ θ < 2π, or along the negative real axis with −π < θ ≤ π. Using the first(last) branch cut the principal square root z {\displaystyle \scriptstyle {\sqrt {z}}} maps z {\displaystyle \scriptstyle z} to the half plane with non-negative imaginary(real) part. The last branch cut is presupposed in mathematical software like Matlab or Scilab. === Roots of unity === The number 1 has n different nth roots in the complex plane, namely 1 , ω , ω 2 , … , ω n − 1 , {\displaystyle 1,\;\omega ,\;\omega ^{2},\;\ldots ,\;\omega ^{n-1},} where ω = e 2 π i n = cos ( 2 π n ) + i sin ( 2 π n ) . {\displaystyle \omega =e^{\frac {2\pi i}{n}}=\cos \left({\frac {2\pi }{n}}\right)+i\sin \left({\frac {2\pi }{n}}\right).} These roots are evenly spaced around the unit circle in the complex plane, at angles which are multiples of 2 π / n {\displaystyle 2\pi /n} . For example, the square roots of unity are 1 and −1, and the fourth roots of unity are 1, i {\displaystyle i} , −1, and − i {\displaystyle -i} . === nth roots === Every complex number has n different nth roots in the complex plane. These are η , η ω , η ω 2 , … , η ω n − 1 , {\displaystyle \eta ,\;\eta \omega ,\;\eta \omega ^{2},\;\ldots ,\;\eta \omega ^{n-1},} where η is a single nth root, and 1, ω, ω2, ... ωn−1 are the nth roots of unity. For example, the four different fourth roots of 2 are 2 4 , i 2 4 , − 2 4 , and − i 2 4 . {\displaystyle {\sqrt[{4}]{2}},\quad i{\sqrt[{4}]{2}},\quad -{\sqrt[{4}]{2}},\quad {\text{and}}\quad -i{\sqrt[{4}]{2}}.} In polar form, a single nth root may be found by the formula r e i θ n = r n ⋅ e i θ / n . {\displaystyle {\sqrt[{n}]{re^{i\theta }}}={\sqrt[{n}]{r}}\cdot e^{i\theta /n}.} Here r is the magnitude (the modulus, also called the absolute value) of the number whose root is to be taken; if the number can be written as a+bi then r = a 2 + b 2 {\displaystyle r={\sqrt {a^{2}+b^{2}}}} . Also, θ {\displaystyle \theta } is the angle formed as one pivots on the origin counterclockwise from the positive horizontal axis to a ray going from the origin to the number; it has the properties that cos θ = a / r , {\displaystyle \cos \theta =a/r,} sin θ = b / r , {\displaystyle \sin \theta =b/r,} and tan θ = b / a . {\displaystyle \tan \theta =b/a.} Thus finding nth roots in the complex plane can be segmented into two steps. First, the magnitude of all the nth roots is the nth root of the magnitude of the original number. Second, the angle between the positive horizontal axis and a ray from the origin to one of the nth roots is θ / n {\displaystyle \theta /n} , where θ {\displaystyle \theta } is the angle defined in the same way for the number whose root is being taken. Furthermore, all n of the nth roots are at equally spaced angles from each other. If n is even, a complex number's nth roots, of which there are an even number, come in additive inverse pairs, so that if a number r1 is one of the nth roots then r2 = −r1 is another. This is because raising the latter's coefficient −1 to the nth power for even n yields 1: that is, (−r1)n = (−1)n × r1n = r1n. As with square roots, the formula above does not define a continuous function over the entire complex plane, but instead has a branch cut at points where θ / n is discontinuous. == Solving polynomials == It was once conjectured that all polynomial equations could be solved algebraically (that is, that all roots of a polynomial could be expressed in terms of a finite number of radicals and elementary operations). However, while this is true for third degree polynomials (cubics) and fourth degree polynomials (quartics), the Abel–Ruffini theorem (1824) shows that this is not true in general when the degree is 5 or greater. For example, the solutions of the equation x 5 = x + 1 {\displaystyle x^{5}=x+1} cannot be expressed in terms of radicals. (cf. quintic equation) == Proof of irrationality for non-perfect nth power x == Assume that x n {\displaystyle {\sqrt[{n}]{x}}} is rational. That is, it can be reduced to a fraction a b {\displaystyle {\frac {a}{b}}} , where a and b are integers without a common factor. This means that x = a n b n {\displaystyle x={\frac {a^{n}}{b^{n}}}} . Since x is an integer, a n {\displaystyle a^{n}} and b n {\displaystyle b^{n}} must share a common factor if b ≠ 1 {\displaystyle b\neq 1} . This means that if b ≠ 1 {\displaystyle b\neq 1} , a n b n {\displaystyle {\frac {a^{n}}{b^{n}}}} is not in simplest form. Thus b should equal 1. Since 1 n = 1 {\displaystyle 1^{n}=1} and n 1 = n {\displaystyle {\frac {n}{1}}=n} , a n b n = a n {\displaystyle {\frac {a^{n}}{b^{n}}}=a^{n}} . This means that x = a n {\displaystyle x=a^{n}} and thus, x n = a {\displaystyle {\sqrt[{n}]{x}}=a} . This implies that x n {\displaystyle {\sqrt[{n}]{x}}} is an integer. Since x is not a perfect nth power, this is impossible. Thus x n {\displaystyle {\sqrt[{n}]{x}}} is irrational. == See also == Geometric mean Twelfth root of two == References == == External links ==
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Wikipedia:Null vector#0
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In mathematics, given a vector space X with an associated quadratic form q, written (X, q), a null vector or isotropic vector is a non-zero element x of X for which q(x) = 0. In the theory of real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space (X, q) which has a null vector is called a pseudo-Euclidean space. The term isotropic vector v when q(v) = 0 has been used in quadratic spaces, and anisotropic space for a quadratic space without null vectors. A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces A and B, X = A + B, where q is positive-definite on A and negative-definite on B. The null cone, or isotropic cone, of X consists of the union of balanced spheres: ⋃ r ≥ 0 { x = a + b : q ( a ) = − q ( b ) = r , a , b ∈ B } . {\displaystyle \bigcup _{r\geq 0}\{x=a+b:q(a)=-q(b)=r,\ \ a,b\in B\}.} The null cone is also the union of the isotropic lines through the origin. == Split algebras == A composition algebra with a null vector is a split algebra. In a composition algebra (A, +, ×, *), the quadratic form is q(x) = x x*. When x is a null vector then there is no multiplicative inverse for x, and since x ≠ 0, A is not a division algebra. In the Cayley–Dickson construction, the split algebras arise in the series bicomplex numbers, biquaternions, and bioctonions, which uses the complex number field C {\displaystyle \mathbb {C} } as the foundation of this doubling construction due to L. E. Dickson (1919). In particular, these algebras have two imaginary units, which commute so their product, when squared, yields +1: ( h i ) 2 = h 2 i 2 = ( − 1 ) ( − 1 ) = + 1. {\displaystyle (hi)^{2}=h^{2}i^{2}=(-1)(-1)=+1.} Then ( 1 + h i ) ( 1 + h i ) ∗ = ( 1 + h i ) ( 1 − h i ) = 1 − ( h i ) 2 = 0 {\displaystyle (1+hi)(1+hi)^{*}=(1+hi)(1-hi)=1-(hi)^{2}=0} so 1 + hi is a null vector. The real subalgebras, split complex numbers, split quaternions, and split-octonions, with their null cones representing the light tracking into and out of 0 ∈ A, suggest spacetime topology. == Examples == The light-like vectors of Minkowski space are null vectors. The four linearly independent biquaternions l = 1 + hi, n = 1 + hj, m = 1 + hk, and m∗ = 1 – hk are null vectors and { l, n, m, m∗ } can serve as a basis for the subspace used to represent spacetime. Null vectors are also used in the Newman–Penrose formalism approach to spacetime manifolds. In the Verma module of a Lie algebra there are null vectors. == References == Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P. (1984). Modern Geometry: Methods and Applications. Translated by Burns, Robert G. Springer. p. 50. ISBN 0-387-90872-2. Shaw, Ronald (1982). Linear Algebra and Group Representations. Vol. 1. Academic Press. p. 151. ISBN 0-12-639201-3. Neville, E. H. (Eric Harold) (1922). Prolegomena to Analytical Geometry in Anisotropic Euclidean Space of Three Dimensions. Cambridge University Press. p. 204.
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Wikipedia:Nullform#0
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In mathematics, a nullform of a vector space acted on linearly by a group is a vector on which all invariants of the group vanish. Nullforms were introduced by Hilbert (1893). (Dieudonné & Carrell 1970, 1971, p.57). == References == Dieudonné, Jean A.; Carrell, James B. (1970), "Invariant theory, old and new", Advances in Mathematics, 4: 1–80, doi:10.1016/0001-8708(70)90015-0, ISSN 0001-8708, MR 0255525 Dieudonné, Jean A.; Carrell, James B. (1971), Invariant theory, old and new, Boston, MA: Academic Press, doi:10.1016/0001-8708(70)90015-0, ISBN 978-0-12-215540-6, MR 0279102 Hilbert, David (1893), "Ueber die vollen Invariantensysteme", Mathematische Annalen, 42 (3), Springer Berlin / Heidelberg: 313–373, doi:10.1007/BF01444162, ISSN 0025-5831
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Wikipedia:Nullspace property#0
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In compressed sensing, the nullspace property gives necessary and sufficient conditions on the reconstruction of sparse signals using the techniques of ℓ 1 {\displaystyle \ell _{1}} -relaxation. The term "nullspace property" originates from Cohen, Dahmen, and DeVore. The nullspace property is often difficult to check in practice, and the restricted isometry property is a more modern condition in the field of compressed sensing. == The technique of == ℓ 1 {\displaystyle \ell _{1}} -relaxation The non-convex ℓ 0 {\displaystyle \ell _{0}} -minimization problem, min x ‖ x ‖ 0 {\displaystyle \min \limits _{x}\|x\|_{0}} subject to A x = b {\displaystyle Ax=b} , is a standard problem in compressed sensing. However, ℓ 0 {\displaystyle \ell _{0}} -minimization is known to be NP-hard in general. As such, the technique of ℓ 1 {\displaystyle \ell _{1}} -relaxation is sometimes employed to circumvent the difficulties of signal reconstruction using the ℓ 0 {\displaystyle \ell _{0}} -norm. In ℓ 1 {\displaystyle \ell _{1}} -relaxation, the ℓ 1 {\displaystyle \ell _{1}} problem, min x ‖ x ‖ 1 {\displaystyle \min \limits _{x}\|x\|_{1}} subject to A x = b {\displaystyle Ax=b} , is solved in place of the ℓ 0 {\displaystyle \ell _{0}} problem. Note that this relaxation is convex and hence amenable to the standard techniques of linear programming - a computationally desirable feature. Naturally we wish to know when ℓ 1 {\displaystyle \ell _{1}} -relaxation will give the same answer as the ℓ 0 {\displaystyle \ell _{0}} problem. The nullspace property is one way to guarantee agreement. == Definition == An m × n {\displaystyle m\times n} complex matrix A {\displaystyle A} has the nullspace property of order s {\displaystyle s} , if for all index sets S {\displaystyle S} with s = | S | ≤ n {\displaystyle s=|S|\leq n} we have that: ‖ η S ‖ 1 < ‖ η S C ‖ 1 {\displaystyle \|\eta _{S}\|_{1}<\|\eta _{S^{C}}\|_{1}} for all η ∈ ker A ∖ { 0 } {\displaystyle \eta \in \ker {A}\setminus \left\{0\right\}} . == Recovery Condition == The following theorem gives necessary and sufficient condition on the recoverability of a given s {\displaystyle s} -sparse vector in C n {\displaystyle \mathbb {C} ^{n}} . The proof of the theorem is a standard one, and the proof supplied here is summarized from Holger Rauhut. Theorem: {\displaystyle {\textbf {Theorem:}}} Let A {\displaystyle A} be a m × n {\displaystyle m\times n} complex matrix. Then every s {\displaystyle s} -sparse signal x ∈ C n {\displaystyle x\in \mathbb {C} ^{n}} is the unique solution to the ℓ 1 {\displaystyle \ell _{1}} -relaxation problem with b = A x {\displaystyle b=Ax} if and only if A {\displaystyle A} satisfies the nullspace property with order s {\displaystyle s} . Proof: {\displaystyle {\textit {Proof:}}} For the forwards direction notice that η S {\displaystyle \eta _{S}} and − η S C {\displaystyle -\eta _{S^{C}}} are distinct vectors with A ( − η S C ) = A ( η S ) {\displaystyle A(-\eta _{S^{C}})=A(\eta _{S})} by the linearity of A {\displaystyle A} , and hence by uniqueness we must have ‖ η S ‖ 1 < ‖ η S C ‖ 1 {\displaystyle \|\eta _{S}\|_{1}<\|\eta _{S^{C}}\|_{1}} as desired. For the backwards direction, let x {\displaystyle x} be s {\displaystyle s} -sparse and z {\displaystyle z} another (not necessary s {\displaystyle s} -sparse) vector such that z ≠ x {\displaystyle z\neq x} and A z = A x {\displaystyle Az=Ax} . Define the (non-zero) vector η = x − z {\displaystyle \eta =x-z} and notice that it lies in the nullspace of A {\displaystyle A} . Call S {\displaystyle S} the support of x {\displaystyle x} , and then the result follows from an elementary application of the triangle inequality: ‖ x ‖ 1 ≤ ‖ x − z S ‖ 1 + ‖ z S ‖ 1 = ‖ η S ‖ 1 + ‖ z S ‖ 1 < ‖ η S C ‖ 1 + ‖ z S ‖ 1 = ‖ − z S C ‖ 1 + ‖ z S ‖ 1 = ‖ z ‖ 1 {\displaystyle \|x\|_{1}\leq \|x-z_{S}\|_{1}+\|z_{S}\|_{1}=\|\eta _{S}\|_{1}+\|z_{S}\|_{1}<\|\eta _{S^{C}}\|_{1}+\|z_{S}\|_{1}=\|-z_{S^{C}}\|_{1}+\|z_{S}\|_{1}=\|z\|_{1}} , establishing the minimality of x {\displaystyle x} . ◻ {\displaystyle \square } == References ==
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