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150-532: Teams consist of 15 members, which usually represent a large geographic region (such as a state) or a large population center (such as a major city). Some schools also field teams. The competition is held in June, on the first Saturday after Memorial Day. In 2022, 120 teams competed with about 1800 students. ARML problems cover a wide variety of mathematical topics including algebra , geometry , number theory , combinatorics , probability , and inequalities . Calculus

300-563: A {\displaystyle a} to object b {\displaystyle b} , and another morphism from object b {\displaystyle b} to object c {\displaystyle c} , then there must also exist one from object a {\displaystyle a} to object c {\displaystyle c} . Composition of morphisms is required to be associative, and there must be an "identity morphism" for every object. Categories are widely used in contemporary mathematics since they provide

450-403: A {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} are usually used for constants and coefficients . The expression 5 x + 3 {\displaystyle 5x+3} is an algebraic expression created by multiplying the number 5 with the variable x {\displaystyle x} and adding

600-746: A 2 x 2 + . . . + a n x n = b {\displaystyle a_{1}x_{1}+a_{2}x_{2}+...+a_{n}x_{n}=b} where a 1 {\displaystyle a_{1}} , a 2 {\displaystyle a_{2}} , ..., a n {\displaystyle a_{n}} and b {\displaystyle b} are constants. Examples are x 1 − 7 x 2 + 3 x 3 = 0 {\displaystyle x_{1}-7x_{2}+3x_{3}=0} and 1 4 x − y = 4 {\textstyle {\frac {1}{4}}x-y=4} . A system of linear equations

750-429: A ∘ a − 1 = a − 1 ∘ a = e {\displaystyle a\circ a^{-1}=a^{-1}\circ a=e} . Every algebraic structure that fulfills these requirements is a group. For example, ⟨ Z , + ⟩ {\displaystyle \langle \mathbb {Z} ,+\rangle } is a group formed by the set of integers together with

900-433: A ∘ b ) ∘ c {\displaystyle (a\circ b)\circ c} is the same as a ∘ ( b ∘ c ) {\displaystyle a\circ (b\circ c)} for all elements. An operation has an identity element or a neutral element if one element e exists that does not change the value of any other element, i.e., if a ∘ e = e ∘

1050-402: A + c a . {\displaystyle (b+c)a=ba+ca.} Moreover, multiplication is associative and has an identity element generally denoted as 1 . Multiplication needs not to be commutative; if it is commutative, one has a commutative ring . The ring of integers ( Z {\displaystyle \mathbb {Z} } ) is one of the simplest commutative rings. A field

1200-437: A = a {\displaystyle a\circ e=e\circ a=a} . An operation has inverse elements if for any element a {\displaystyle a} there exists a reciprocal element a − 1 {\displaystyle a^{-1}} that undoes a {\displaystyle a} . If an element operates on its inverse then the result is the neutral element e , expressed formally as

1350-652: A Lie algebra or an associative algebra . The word algebra comes from the Arabic term الجبر ( al-jabr ), which originally referred to the surgical treatment of bonesetting . In the 9th century, the term received a mathematical meaning when the Persian mathematician Muhammad ibn Musa al-Khwarizmi employed it to describe a method of solving equations and used it in the title of a treatise on algebra, al-Kitāb al-Mukhtaṣar fī Ḥisāb al-Jabr wal-Muqābalah [ The Compendious Book on Calculation by Completion and Balancing ] which

1500-625: A healthy diet are also recommended. Supportive care in severe cases includes treatment to relieve symptoms , fluid therapy , oxygen support and prone positioning , and medications or devices to support other affected vital organs. More severe cases may need treatment in hospital. In those with low oxygen levels, use of the glucocorticoid dexamethasone is recommended to reduce mortality. Noninvasive ventilation and, ultimately, admission to an intensive care unit for mechanical ventilation may be required to support breathing. Extracorporeal membrane oxygenation (ECMO) has been used to address

1650-623: A set of mathematical objects together with one or several operations defined on that set. It is a generalization of elementary and linear algebra, since it allows mathematical objects other than numbers and non-arithmetic operations. It distinguishes between different types of algebraic structures, such as groups , rings , and fields , based on the number of operations they use and the laws they follow . Universal algebra and category theory provide general frameworks to investigate abstract patterns that characterize different classes of algebraic structures. Algebraic methods were first studied in

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1800-404: A theory of equations to cover diverse types of algebraic operations and structures. Algebra is relevant to many branches of mathematics, such as geometry, topology , number theory , and calculus , and other fields of inquiry, like logic and the empirical sciences . Algebra is the branch of mathematics that studies algebraic structures and the operations they use. An algebraic structure

1950-615: A variant of concern (VoC) or a variant of interest (VoI). Many of these variants have shared the more infectious D614G . As of May 2023, the WHO had downgraded all variants of concern to previously circulating as these were no longer detected in new infections. Sub-lineages of the Omicron variant (BA.1 – BA.5) were considered separate VoCs by the WHO until they were downgraded in March 2023 as no longer widely circulating. As of 24 September 2024 ,

2100-493: A comprehensive international study. Such deaths include deaths due to healthcare capacity constraints and priorities, as well as reluctance to seek care (to avoid possible infection). Further research may help distinguish the proportions directly caused by COVID-19 from those caused by indirect consequences of the pandemic. In May 2022, the WHO estimated the number of excess deaths by the end of 2021 to be 14.9 million compared to 5.4 million reported COVID-19 deaths, with

2250-404: A generalization of arithmetic . Arithmetic studies operations like addition, subtraction , multiplication, and division , in a particular domain of numbers, such as the real numbers. Elementary algebra constitutes the first level of abstraction. Like arithmetic, it restricts itself to specific types of numbers and operations. It generalizes these operations by allowing indefinite quantities in

2400-461: A key turning point in the history of algebra and consider what came before it as the prehistory of algebra because it lacked the abstract nature based on symbolic manipulation. In the 17th and 18th centuries, many attempts were made to find general solutions to polynomials of degree five and higher. All of them failed. At the end of the 18th century, the German mathematician Carl Friedrich Gauss proved

2550-590: A large number of people". During the COVID-19 pandemic, as with other pandemics, the meaning of this term has been challenged. The end of a pandemic or other epidemic only rarely involves the total disappearance of a disease, and historically, much less attention has been given to defining the ends of epidemics than their beginnings. The ends of particular epidemics have been defined in a variety of ways, differing according to academic field, and differently based on location and social group. An epidemic's end can be considered

2700-467: A large part of linear algebra. A vector space is an algebraic structure formed by a set with an addition that makes it an abelian group and a scalar multiplication that is compatible with addition (see vector space for details). A linear map is a function between vector spaces that is compatible with addition and scalar multiplication. In the case of finite-dimensional vector spaces , vectors and linear maps can be represented by matrices. It follows that

2850-448: A median range of 0.24% to 1.49%. IFRs rise as a function of age (from 0.002% at age 10 and 0.01% at age 25, to 0.4% at age 55, 1.4% at age 65, 4.6% at age 75, and 15% at age 85). These rates vary by a factor of ≈10,000 across the age groups. For comparison, the IFR for middle-aged adults is two orders of magnitude higher than the annualised risk of a fatal automobile accident and much higher than

3000-412: A positive degree can be factorized into linear polynomials. This theorem was proved at the beginning of the 19th century, but this does not close the problem since the theorem does not provide any way for computing the solutions. Linear algebra starts with the study systems of linear equations . An equation is linear if it can be expressed in the form a 1 x 1 +

3150-815: A random sample have consistently found that total infections considerably exceed the reported case counts. Many countries, early on, had official policies to not test those with only mild symptoms. The strongest risk factors for severe illness are obesity, complications of diabetes , anxiety disorders, and the total number of conditions. During the start of the COVID-19 pandemic it was not clear whether young people were less likely to be infected, or less likely to develop symptoms and be tested. A retrospective cohort study in China found that children and adults were just as likely to be infected. Among more thorough studies, preliminary results from 9 April 2020 found that in Gangelt ,

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3300-428: A second-degree polynomial equation of the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} is given by the quadratic formula x = − b ± b 2 − 4 a c   2 a . {\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac\ }}}{2a}}.} Solutions for

3450-447: A similar way, if one knows the value of one variable one may be able to use it to determine the value of other variables. Algebraic equations can be interpreted geometrically to describe spatial figures in the form of a graph . To do so, the different variables in the equation are understood as coordinates and the values that solve the equation are interpreted as points of a graph. For example, if x {\displaystyle x}

3600-457: A social phenomenon, not just a biological one. Time reported in March 2024 that expert opinions differ on whether or not COVID-19 is considered endemic or pandemic, and that the WHO continued to call the disease a pandemic on its website. During the initial outbreak in Wuhan , the virus and disease were commonly referred to as "coronavirus", "Wuhan coronavirus", "the coronavirus outbreak" and

3750-435: A statement formed by comparing two expressions, saying that they are equal. This can be expressed using the equals sign ( = {\displaystyle =} ), as in 5 x 2 + 6 x = 3 y + 4 {\displaystyle 5x^{2}+6x=3y+4} . Inequations involve a different type of comparison, saying that the two sides are different. This can be expressed using symbols such as

3900-468: A student answers a tiebreaker question correctly, they have finished the tiebreaker. Students are then ranked by individual round score, followed by the time to answer the first tiebreaker question correctly, then the time to answer the second tiebreaker (if applicable), then the third. Students are given 10 minutes to answer the first tiebreaker question, and 6 minutes to answer each of the second and third tiebreaker questions. In recent years, there has been

4050-453: A super relay, where two groups of seven team members (fourteen in all) both work to give a correct answer to the fifteenth team member. That last team member substitutes two answers into his problem. For logistical reasons, the Super Relay has never counted towards the team score. It was instituted as a "filler" while scores are tabulated. Candies and other goodies are sometimes rewards for

4200-451: A test. Conversely, deaths of people who died from underlying conditions following a positive test may be included. Countries such as Belgium include deaths from suspected cases, including those without a test, thereby increasing counts. Official death counts have been claimed to underreport the actual death toll, because excess mortality (the number of deaths in a period compared to a long-term average) data show an increase in deaths that

4350-527: A unifying framework to describe and analyze many fundamental mathematical concepts. For example, sets can be described with the category of sets , and any group can be regarded as the morphisms of a category with just one object. The origin of algebra lies in attempts to solve mathematical problems involving arithmetic calculations and unknown quantities. These developments happened in the ancient period in Babylonia , Egypt , Greece , China , and India . One of

4500-400: A whole is zero if and only if one of its factors is zero, i.e., if x {\displaystyle x} is either −2 or 5. Before the 19th century, much of algebra was devoted to polynomial equations , that is equations obtained by equating a polynomial to zero. The first attempts for solving polynomial equations was to express the solutions in terms of n th roots . The solution of

4650-397: Is a commutative group under addition: the addition of the ring is associative, commutative, and has an identity element and inverse elements. The multiplication is associative and distributive with respect to addition; that is, a ( b + c ) = a b + a c {\displaystyle a(b+c)=ab+ac} and ( b + c ) a = b

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4800-438: Is a commutative ring such that ⁠ 1 ≠ 0 {\displaystyle 1\neq 0} ⁠ and each nonzero element has a multiplicative inverse . The ring of integers does not form a field because it lacks multiplicative inverses. For example, the multiplicative inverse of 7 {\displaystyle 7} is 1 7 {\displaystyle {\tfrac {1}{7}}} , which

4950-475: Is a function from the underlying set of one algebraic structure to the underlying set of another algebraic structure that preserves certain structural characteristics. If the two algebraic structures use binary operations and have the form ⟨ A , ∘ ⟩ {\displaystyle \langle A,\circ \rangle } and ⟨ B , ⋆ ⟩ {\displaystyle \langle B,\star \rangle } then

5100-504: Is a method used to simplify polynomials, making it easier to analyze them and determine the values for which they evaluate to zero . Factorization consists in rewriting a polynomial as a product of several factors. For example, the polynomial x 2 − 3 x − 10 {\displaystyle x^{2}-3x-10} can be factorized as ( x + 2 ) ( x − 5 ) {\displaystyle (x+2)(x-5)} . The polynomial as

5250-415: Is a non-empty set of mathematical objects , such as the integers , together with algebraic operations defined on that set, like addition and multiplication . Algebra explores the laws, general characteristics, and types of algebraic structures. Within certain algebraic structures, it examines the use of variables in equations and how to manipulate these equations. Algebra is often understood as

5400-487: Is a polynomial with one term while two- and three-term polynomials are called binomials and trinomials. The degree of a polynomial is the maximal value (among its terms) of the sum of the exponents of the variables (4 in the above example). Polynomials of degree one are called linear polynomials . Linear algebra studies systems of linear polynomials. A polynomial is said to be univariate or multivariate , depending on whether it uses one or more variables. Factorization

5550-941: Is a set of linear equations for which one is interested in common solutions. Matrices are rectangular arrays of values that have been originally introduced for having a compact and synthetic notation for systems of linear equations For example, the system of equations 9 x 1 + 3 x 2 − 13 x 3 = 0 2.3 x 1 + 7 x 3 = 9 − 5 x 1 − 17 x 2 = − 3 {\displaystyle {\begin{aligned}9x_{1}+3x_{2}-13x_{3}&=0\\2.3x_{1}+7x_{3}&=9\\-5x_{1}-17x_{2}&=-3\end{aligned}}} can be written as A X = B , {\displaystyle AX=B,} where A , B {\displaystyle A,B} and C {\displaystyle C} are

5700-414: Is an expression consisting of one or more terms that are added or subtracted from each other, like x 4 + 3 x y 2 + 5 x 3 − 1 {\displaystyle x^{4}+3xy^{2}+5x^{3}-1} . Each term is either a constant, a variable, or a product of a constant and variables. Each variable can be raised to a positive-integer power. A monomial

5850-629: Is applied to one side of an equation also needs to be done to the other side. For example, if one subtracts 5 from the left side of an equation one also needs to subtract 5 from the right side to balance both sides. The goal of these steps is usually to isolate the variable one is interested in on one side, a process known as solving the equation for that variable. For example, the equation x − 7 = 4 {\displaystyle x-7=4} can be solved for x {\displaystyle x} by adding 7 to both sides, which isolates x {\displaystyle x} on

6000-518: Is associated with COVID-19 and is reported in as many as 88% of cases. The disease is mainly transmitted via the respiratory route when people inhale droplets and small airborne particles (that form an aerosol ) that infected people exhale as they breathe, talk, cough, sneeze, or sing. Infected people are more likely to transmit COVID-19 when they are physically close to other non-infected individuals. However, infection can occur over longer distances, particularly indoors. SARS‑CoV‑2 belongs to

6150-503: Is discouraged altogether by the WHO, due to limited evidence of its efficacy. The severity of COVID-19 varies. It may take a mild course with few or no symptoms, resembling other common upper respiratory diseases such as the common cold . In 3–4% of cases (7.4% for those over age 65) symptoms are severe enough to cause hospitalisation. Mild cases typically recover within two weeks, while those with severe or critical diseases may take three to six weeks to recover. Among those who have died,

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6300-439: Is intended to provide acquired immunity against severe acute respiratory syndrome coronavirus 2 (SARS‑CoV‑2), the virus that causes coronavirus disease 2019 ( COVID-19 ). Prior to the COVID-19 pandemic, an established body of knowledge existed about the structure and function of coronaviruses causing diseases like severe acute respiratory syndrome (SARS) and Middle East respiratory syndrome (MERS). This knowledge accelerated

6450-408: Is no solution since they never intersect. If two equations are not independent then they describe the same line, meaning that every solution of one equation is also a solution of the other equation. These relations make it possible to seek solutions graphically by plotting the equations and determining where they intersect. The same principles also apply to systems of equations with more variables, with

6600-482: Is not an integer. The rational numbers , the real numbers , and the complex numbers each form a field with the operations of addition and multiplication. Ring theory is the study of rings, exploring concepts such as subrings , quotient rings , polynomial rings , and ideals as well as theorems such as Hilbert's basis theorem . Field theory is concerned with fields, examining field extensions , algebraic closures , and finite fields . Galois theory explores

6750-485: Is not explained by COVID-19 deaths alone. Using such data, estimates of the true number of deaths from COVID-19 worldwide have included a range from 18.2 to 33.5 million (≈27.4 million) by 18 November 2023 by The Economist , as well as over 18.5 million by 1 April 2023 by the Institute for Health Metrics and Evaluation and ≈18.2 million (earlier) deaths between 1 January 2020, and 31 December 2021, by

6900-426: Is not required to successfully complete any problem, but it may facilitate solving the problem more quickly or efficiently. While part of the competition is short-answer based, there is a cooperative team round, and a proof-based power question (also completed as a team). ARML problems are harder than most high school mathematics competitions . The contest is sponsored by D. E. Shaw & Co. Contest supporters are

7050-400: Is set to zero in the equation y = 0.5 x − 1 {\displaystyle y=0.5x-1} , then y {\displaystyle y} must be −1 for the equation to be true. This means that the ( x , y ) {\displaystyle (x,y)} -pair ( 0 , − 1 ) {\displaystyle (0,-1)} is part of

7200-403: Is the identity matrix . Then, multiplying on the left both members of the above matrix equation by A − 1 , {\displaystyle A^{-1},} one gets the solution of the system of linear equations as X = A − 1 B . {\displaystyle X=A^{-1}B.} Methods of solving systems of linear equations range from

7350-414: Is the application of group theory to analyze graphs and symmetries. The insights of algebra are also relevant to calculus, which uses mathematical expressions to examine rates of change and accumulation . It relies on algebra, for instance, to understand how these expressions can be transformed and what role variables play in them. Algebraic logic employs the methods of algebra to describe and analyze

7500-425: Is the case because the sum of two even numbers is again an even number. But the set of odd integers together with addition is not a subalgebra because it is not closed: adding two odd numbers produces an even number, which is not part of the chosen subset. Universal algebra is the study of algebraic structures in general. As part of its general perspective, it is not concerned with the specific elements that make up

7650-421: Is the process of applying algebraic methods and principles to other branches of mathematics , such as geometry , topology , number theory , and calculus . It happens by employing symbols in the form of variables to express mathematical insights on a more general level, allowing mathematicians to develop formal models describing how objects interact and relate to each other. One application, found in geometry,

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7800-452: Is the ratio of deaths to diagnoses. This metric can be misleading because of the delay between symptom onset and death and because testing focuses on symptomatic individuals. Based on Johns Hopkins University statistics, the global CFR was 1.02 percent (6,881,955 deaths for 676,609,955 cases) as of 10 March 2023. The number varies by region and has generally declined over time. Several variants have been named by WHO and labelled as

7950-406: Is the seventh known coronavirus to infect people, after 229E , NL63 , OC43 , HKU1 , MERS-CoV , and the original SARS-CoV . The standard method of testing for presence of SARS-CoV-2 is a nucleic acid test , which detects the presence of viral RNA fragments. As these tests detect RNA but not infectious virus, its "ability to determine duration of infectivity of patients is limited". The test

8100-472: Is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of addition , subtraction , multiplication , division , exponentiation , extraction of roots , and logarithm . For example, the operation of addition combines two numbers, called the addends, into a third number, called the sum, as in 2 + 5 = 7 {\displaystyle 2+5=7} . Elementary algebra relies on

8250-718: Is the use of algebraic statements to describe geometric figures. For example, the equation y = 3 x − 7 {\displaystyle y=3x-7} describes a line in two-dimensional space while the equation x 2 + y 2 + z 2 = 1 {\displaystyle x^{2}+y^{2}+z^{2}=1} corresponds to a sphere in three-dimensional space. Of special interest to algebraic geometry are algebraic varieties , which are solutions to systems of polynomial equations that can be used to describe more complex geometric figures. Algebraic reasoning can also solve geometric problems. For example, one can determine whether and where

8400-466: Is true for all elements of the underlying set. For example, commutativity is a universal equation that states that a ∘ b {\displaystyle a\circ b} is identical to b ∘ a {\displaystyle b\circ a} for all elements. A variety is a class of all algebraic structures that satisfy certain identities. For example, if two algebraic structures satisfy commutativity then they are both part of

8550-446: Is true if x {\displaystyle x} is either 2 or −2 and false otherwise. Equations with variables can be divided into identity equations and conditional equations. Identity equations are true for all values that can be assigned to the variables, such as the equation 2 x + 5 x = 7 x {\displaystyle 2x+5x=7x} . Conditional equations are only true for some values. For example,

8700-614: Is typically done on respiratory samples obtained by a nasopharyngeal swab ; however, a nasal swab or sputum sample may also be used. The WHO has published several testing protocols for the disease. Preventive measures to reduce the chances of infection include getting vaccinated, staying at home or spending more time outdoors, avoiding crowded places, keeping distance from others, wearing a mask in public, ventilating indoor spaces, managing potential exposure durations, washing hands with soap and water often and for at least twenty seconds, practising good respiratory hygiene, and avoiding touching

8850-599: The American Mathematical Society , Mu Alpha Theta (the National Mathematics Honor Society for High School and Two-Year College students), Star League, Penguin Books , and Princeton University Press . The competition consists of four formal events: The teams are scored based on the number of points they attained with the maximum being 300 points. Team score ties are broken by first considering

9000-501: The Huanan Seafood Wholesale Market there, but it is possible that human-to-human transmission began earlier. Molecular clock analysis suggests that the first cases were likely to have been between October and November 2019. The scientific consensus is that the virus is most likely of a zoonotic origin, from bats or another closely related mammal. While other explanations such as speculations that SARS-CoV-2

9150-407: The Institute for Health Metrics and Evaluation estimating the true number of cases as of early 2022 to be in the billions. One measure that public health officials and policymakers have used to monitor the pandemic and guide decision-making is the test positivity rate ("percent positive"). According to Johns Hopkins in 2020, one benchmark for a "too high" per cent positive is 5%, which was used by

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9300-414: The ancient period to solve specific problems in fields like geometry . Subsequent mathematicians examined general techniques to solve equations independent of their specific applications. They described equations and their solutions using words and abbreviations until the 16th and 17th centuries, when a rigorous symbolic formalism was developed. In the mid-19th century, the scope of algebra broadened beyond

9450-597: The basic reproduction number (R 0 ) for COVID-19 in January 2020 were between 1.4 and 2.5, but a subsequent analysis claimed that it may be about 5.7 (with a 95 per cent confidence interval of 3.8 to 8.9). In December 2021, the number of cases continued to climb due to several factors, including new COVID-19 variants. As of that 28   December, 282,790,822 individuals worldwide had been confirmed as infected. As of 14 April 2022 , over 500 million cases were confirmed globally. Most cases are unconfirmed, with

9600-474: The coronavirus pandemic and COVID pandemic ), caused by severe acute respiratory syndrome coronavirus 2 ( SARS-CoV-2 ), began with an outbreak of COVID-19 in Wuhan , China, in December 2019. It spread to other areas of Asia, and then worldwide in early 2020. The World Health Organization (WHO) declared the outbreak a public health emergency of international concern (PHEIC) on 30 January 2020, and assessed

9750-547: The difference of two squares method and later in Euclid's Elements . In the 3rd century CE, Diophantus provided a detailed treatment of how to solve algebraic equations in a series of books called Arithmetica . He was the first to experiment with symbolic notation to express polynomials. Diophantus's work influenced Arab development of algebra with many of his methods reflected in the concepts and techniques used in medieval Arabic algebra. In ancient China, The Nine Chapters on

9900-525: The fundamental theorem of algebra , which describes the existence of zeros of polynomials of any degree without providing a general solution. At the beginning of the 19th century, the Italian mathematician Paolo Ruffini and the Norwegian mathematician Niels Henrik Abel were able to show that no general solution exists for polynomials of degree five and higher. In response to and shortly after their findings,

10050-602: The fundamental theorem of finite abelian groups and the Feit–Thompson theorem . The latter was a key early step in one of the most important mathematical achievements of the 20th century: the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete classification of finite simple groups . A ring is an algebraic structure with two operations that work similarly to addition and multiplication of numbers and are named and generally denoted similarly. A ring

10200-461: The less-than sign ( < {\displaystyle <} ), the greater-than sign ( > {\displaystyle >} ), and the inequality sign ( ≠ {\displaystyle \neq } ). Unlike other expressions, statements can be true or false and their truth value usually depends on the values of the variables. For example, the statement x 2 = 4 {\displaystyle x^{2}=4}

10350-450: The variants of interest as specified by the WHO are BA.2.86 and JN.1, and the variants under monitoring are JN.1.7, KP.2, KP.3, KP.3.1.1, JN.1.18, LB.1, and XEC. Symptoms of COVID-19 are variable, ranging from mild symptoms to severe illness. Common symptoms include headache, loss of smell and taste , nasal congestion and runny nose , cough, muscle pain , sore throat , fever, diarrhoea , and breathing difficulties . People with

10500-429: The "Wuhan coronavirus outbreak", with the disease sometimes called "Wuhan pneumonia ". In January 2020, the WHO recommended 2019-nCoV and 2019-nCoV acute respiratory disease as interim names for the virus and disease per 2015 international guidelines against using geographical locations (e.g. Wuhan, China), animal species, or groups of people in disease and virus names in part to prevent social stigma . WHO finalised

10650-653: The 12th century further refined Brahmagupta's methods and concepts. In 1247, the Chinese mathematician Qin Jiushao wrote the Mathematical Treatise in Nine Sections , which includes an algorithm for the numerical evaluation of polynomials , including polynomials of higher degrees. The Italian mathematician Fibonacci brought al-Khwarizmi's ideas and techniques to Europe in books including his Liber Abaci . In 1545,

10800-528: The 1930s, the American mathematician Garrett Birkhoff expanded these ideas and developed many of the foundational concepts of this field. The invention of universal algebra led to the emergence of various new areas focused on the algebraization of mathematics—that is, the application of algebraic methods to other branches of mathematics. Topological algebra arose in the early 20th century, studying algebraic structures such as topological groups and Lie groups . In

10950-464: The 1940s and 50s, homological algebra emerged, employing algebraic techniques to study homology . Around the same time, category theory was developed and has since played a key role in the foundations of mathematics . Other developments were the formulation of model theory and the study of free algebras . The influence of algebra is wide-reaching, both within mathematics and in its applications to other fields. The algebraization of mathematics

11100-402: The 9th century and the Persian mathematician Omar Khayyam in the 11th and 12th centuries. In India, Brahmagupta investigated how to solve quadratic equations and systems of equations with several variables in the 7th century CE. Among his innovations were the use of zero and negative numbers in algebraic equations. The Indian mathematicians Mahāvīra in the 9th century and Bhāskara II in

11250-586: The Atlantic Regions Mathematics League was founded, the competition was held at a single eastern site that changed from year to year: After 1983, the coordinators decided to keep the competition at Penn State University. ARML expanded to two sites in the late 1980s and to three sites in 1995. In 2008, ARML added a fourth site at the University of Georgia in Athens to better accommodate students in

11400-481: The French mathematician Évariste Galois developed what came later to be known as Galois theory , which offered a more in-depth analysis of the solutions of polynomials while also laying the foundation of group theory . Mathematicians soon realized the relevance of group theory to other fields and applied it to disciplines like geometry and number theory. Starting in the mid-19th century, interest in algebra shifted from

11550-598: The German mathematicians David Hilbert , Ernst Steinitz , and Emmy Noether as well as the Austrian mathematician Emil Artin . They researched different forms of algebraic structures and categorized them based on their underlying axioms into types, like groups, rings, and fields. The idea of the even more general approach associated with universal algebra was conceived by the English mathematician Alfred North Whitehead in his 1898 book A Treatise on Universal Algebra . Starting in

11700-635: The Great Plains Math League. The New York State Mathematics League held its first competition in 1973, a competition intended for New York state teams. A team from Massachusetts asked to participate in the 1974 NYSML competition, and it took first place. This led to the creation of the Atlantic Regions Mathematics League in 1976, which became the American Regions Mathematics League in 1984. When

11850-592: The Italian polymath Gerolamo Cardano published his book Ars Magna , which covered many topics in algebra, discussed imaginary numbers , and was the first to present general methods for solving cubic and quartic equations . In the 16th and 17th centuries, the French mathematicians François Viète and René Descartes introduced letters and symbols to denote variables and operations, making it possible to express equations in an abstract and concise manner. Their predecessors had relied on verbal descriptions of problems and solutions. Some historians see this development as

12000-561: The Mathematical Art , a book composed over the period spanning from the 10th century BCE to the 2nd century CE, explored various techniques for solving algebraic equations, including the use of matrix-like constructs. There is no unanimity as to whether these early developments are part of algebra or only precursors. They offered solutions to algebraic problems but did not conceive them in an abstract and general manner, focusing instead on specific cases and applications. This changed with

12150-454: The Persian mathematician al-Khwarizmi , who published his The Compendious Book on Calculation by Completion and Balancing in 825 CE. It presents the first detailed treatment of general methods that can be used to manipulate linear and quadratic equations by "reducing" and "balancing" both sides. Other influential contributions to algebra came from the Arab mathematician Thābit ibn Qurra also in

12300-700: The Southeast, which moved to the University of Alabama in Huntsville in 2020. The 2006 competition saw significant expansion of about 25% more participants than ever before. Attendance at the western site, UNLV , nearly doubled. Due to the COVID-19 pandemic , the ARML competition was not held in 2020, and it was only held virtually in 2021. Algebra Algebra is the branch of mathematics that studies certain abstract systems , known as algebraic structures , and

12450-636: The WHO in the past. As of 10 March 2023, more than 6.88   million deaths had been attributed to COVID-19. The first confirmed death was in Wuhan on 9 January 2020. These numbers vary by region and over time, influenced by testing volume, healthcare system quality, treatment options, government response, time since the initial outbreak, and population characteristics, such as age, sex, and overall health. Multiple measures are used to quantify mortality. Official death counts typically include people who died after testing positive. Such counts exclude deaths without

12600-419: The WHO, US, European Union, and Australia. On 12 November 2022, the WHO released its Global Vaccine Market Report. The report indicated that "inequitable distribution is not unique to COVID-19 vaccines"; countries that are not economically strong struggle to obtain vaccines. On 14 November 2022, the first inhalable vaccine was introduced, developed by Chinese biopharmaceutical company CanSino Biologics , in

12750-401: The addition of numbers. While elementary algebra and linear algebra work within the confines of particular algebraic structures, abstract algebra takes a more general approach that compares how algebraic structures differ from each other and what types of algebraic structures there are, such as groups , rings , and fields . The key difference between these types of algebraic structures lies in

12900-448: The broad family of viruses known as coronaviruses . It is a positive-sense single-stranded RNA (+ssRNA) virus, with a single linear RNA segment. Coronaviruses infect humans, other mammals, including livestock and companion animals, and avian species. Human coronaviruses are capable of causing illnesses ranging from the common cold to more severe diseases such as Middle East respiratory syndrome (MERS, fatality rate ≈34%). SARS-CoV-2

13050-630: The centre of a major infection cluster in Germany, 15 per cent of a population sample tested positive for antibodies . Screening for COVID-19 in pregnant women in New York City , and blood donors in the Netherlands, found rates of positive antibody tests that indicated more infections than reported. Seroprevalence -based estimates are conservative as some studies show that persons with mild symptoms do not have detectable antibodies. Initial estimates of

13200-443: The characteristics of algebraic structures in general. The term "algebra" is sometimes used in a more narrow sense to refer only to elementary algebra or only to abstract algebra. When used as a countable noun , an algebra is a specific type of algebraic structure that involves a vector space equipped with a certain type of binary operation . Depending on the context, "algebra" can also refer to other algebraic structures, like

13350-746: The city of Shanghai, China. For the first two years of the pandemic, no specific and effective treatment or cure was available. In 2021, the European Medicines Agency's (EMA) Committee for Medicinal Products for Human Use (CHMP) approved the oral antiviral protease inhibitor , Paxlovid (nirmatrelvir plus the HIV antiviral ritonavir ), to treat adult patients. FDA later gave it an EUA. Most cases of COVID-19 are mild. In these, supportive care includes medication such as paracetamol or NSAIDs to relieve symptoms (fever, body aches, cough), adequate intake of oral fluids and rest. Good personal hygiene and

13500-416: The corresponding variety. Category theory examines how mathematical objects are related to each other using the concept of categories . A category is a collection of objects together with a collection of so-called morphisms or "arrows" between those objects. These two collections must satisfy certain conditions. For example, morphisms can be joined, or composed : if there exists a morphism from object

13650-480: The cumulative number of infected individuals (including asymptomatic and undiagnosed infections and excluding vaccinated infected individuals). It is expressed in percentage points. Other studies refer to this metric as the infection fatality risk . In November 2020, a review article in Nature reported estimates of population-weighted IFRs for various countries, excluding deaths in elderly care facilities, and found

13800-593: The degrees 3 and 4 are given by the cubic and quartic formulas. There are no general solutions for higher degrees, as proven in the 19th century by the so-called Abel–Ruffini theorem . Even when general solutions do not exist, approximate solutions can be found by numerical tools like the Newton–Raphson method . The fundamental theorem of algebra asserts that every univariate polynomial equation of positive degree with real or complex coefficients has at least one complex solution. Consequently, every polynomial of

13950-441: The development of various vaccine platforms during early 2020. The initial focus of SARS-CoV-2 vaccines was on preventing symptomatic and severe illness. The COVID-19 vaccines are widely credited for their role in reducing the severity and death caused by COVID-19. As of March 2023, more than 5.5 billion people had received one or more doses (11.8 billion in total) in over 197 countries. The Oxford-AstraZeneca vaccine

14100-455: The difference being that the equations do not describe lines but higher dimensional figures. For instance, equations with three variables correspond to planes in three-dimensional space , and the points where all planes intersect solve the system of equations. Abstract algebra, also called modern algebra, is the study of algebraic structures . An algebraic structure is a framework for understanding operations on mathematical objects , like

14250-469: The distributive property. For statements with several variables, substitution is a common technique to replace one variable with an equivalent expression that does not use this variable. For example, if one knows that y = 3 x {\displaystyle y=3x} then one can simplify the expression 7 x y {\displaystyle 7xy} to arrive at 21 x 2 {\displaystyle 21x^{2}} . In

14400-496: The earliest documents on algebraic problems is the Rhind Mathematical Papyrus from ancient Egypt, which was written around 1650 BCE. It discusses solutions to linear equations , as expressed in problems like "A quantity; its fourth is added to it. It becomes fifteen. What is the quantity?" Babylonian clay tablets from around the same time explain methods to solve linear and quadratic polynomial equations , such as

14550-403: The elements of the two algebraic structures. This implies that every element of the first algebraic structure is mapped to one unique element in the second structure without any unmapped elements in the second structure. Another tool of comparison is the relation between an algebraic structure and its subalgebra . The algebraic structure and its subalgebra use the same operations, which follow

14700-404: The equation x + 4 = 9 {\displaystyle x+4=9} is only true if x {\displaystyle x} is 5. The main goal of elementary algebra is to determine the values for which a statement is true. This can be achieved by transforming and manipulating statements according to certain rules. A key principle guiding this process is that whatever operation

14850-612: The existence of loops or holes in them. Number theory is concerned with the properties of and relations between integers. Algebraic number theory applies algebraic methods and principles to this field of inquiry. Examples are the use of algebraic expressions to describe general laws, like Fermat's Last Theorem , and of algebraic structures to analyze the behavior of numbers, such as the ring of integers . The related field of combinatorics uses algebraic techniques to solve problems related to counting, arrangement, and combination of discrete objects. An example in algebraic combinatorics

15000-513: The eyes, nose, or mouth with unwashed hands. Those diagnosed with COVID-19 or who believe they may be infected are advised by healthcare authorities to stay home except to get medical care, call ahead before visiting a healthcare provider, wear a face mask before entering the healthcare provider's office and when in any room or vehicle with another person, cover coughs and sneezes with a tissue, regularly wash hands with soap and water and avoid sharing personal household items. A COVID-19 vaccine

15150-429: The form of variables in addition to numbers. A higher level of abstraction is found in abstract algebra , which is not limited to a particular domain and examines algebraic structures such as groups and rings . It extends beyond typical arithmetic operations by also covering other types of operations. Universal algebra is still more abstract in that it is not interested in specific algebraic structures but investigates

15300-438: The function h : A → B {\displaystyle h:A\to B} is a homomorphism if it fulfills the following requirement: h ( x ∘ y ) = h ( x ) ⋆ h ( y ) {\displaystyle h(x\circ y)=h(x)\star h(y)} . The existence of a homomorphism reveals that the operation ⋆ {\displaystyle \star } in

15450-474: The general public beginning in December 2020, made available through government and international programs such as COVAX , aiming to provide vaccine equity . Treatments include novel antiviral drugs and symptom control. Common mitigation measures during the public health emergency included travel restrictions , lockdowns , business restrictions and closures, workplace hazard controls , mask mandates , quarantines, testing systems, and contact tracing of

15600-413: The graph of the equation. The ( x , y ) {\displaystyle (x,y)} -pair ( 0 , 7 ) {\displaystyle (0,7)} , by contrast, does not solve the equation and is therefore not part of the graph. The graph encompasses the totality of ( x , y ) {\displaystyle (x,y)} -pairs that solve the equation. A polynomial

15750-657: The infected. The pandemic caused severe social and economic disruption around the world, including the largest global recession since the Great Depression . Widespread supply shortages , including food shortages , were caused by supply chain disruptions and panic buying . Reduced human activity led to an unprecedented temporary decrease in pollution . Educational institutions and public areas were partially or fully closed in many jurisdictions, and many events were cancelled or postponed during 2020 and 2021. Telework became much more common for white-collar workers as

15900-495: The introductory, like substitution and elimination, to more advanced techniques using matrices, such as Cramer's rule , the Gaussian elimination , and LU decomposition . Some systems of equations are inconsistent , meaning that no solutions exist because the equations contradict each other. Consistent systems have either one unique solution or an infinite number of solutions. The study of vector spaces and linear maps form

16050-484: The issue of respiratory failure. Existing drugs such as hydroxychloroquine , lopinavir/ritonavir , and ivermectin are not recommended by US or European health authorities, as there is no good evidence they have any useful effect. The antiviral remdesivir is available in the US, Canada, Australia, and several other countries, with varying restrictions; however, it is not recommended for use with mechanical ventilation, and

16200-607: The left side and results in the equation x = 11 {\displaystyle x=11} . There are many other techniques used to solve equations. Simplification is employed to replace a complicated expression with an equivalent simpler one. For example, the expression 7 x − 3 x {\displaystyle 7x-3x} can be replaced with the expression 4 x {\displaystyle 4x} since 7 x − 3 x = ( 7 − 3 ) x = 4 x {\displaystyle 7x-3x=(7-3)x=4x} by

16350-620: The line described by y = x + 1 {\displaystyle y=x+1} intersects with the circle described by x 2 + y 2 = 25 {\displaystyle x^{2}+y^{2}=25} by solving the system of equations made up of these two equations. Topology studies the properties of geometric figures or topological spaces that are preserved under operations of continuous deformation . Algebraic topology relies on algebraic theories such as group theory to classify topological spaces. For example, homotopy groups classify topological spaces based on

16500-426: The linear map to the basis vectors. Systems of equations can be interpreted as geometric figures. For systems with two variables, each equation represents a line in two-dimensional space . The point where the two lines intersect is the solution of the full system because this is the only point that solves both the first and the second equation. For inconsistent systems, the two lines run parallel, meaning that there

16650-472: The lowercase letters x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} represent variables. In some cases, subscripts are added to distinguish variables, as in x 1 {\displaystyle x_{1}} , x 2 {\displaystyle x_{2}} , and x 3 {\displaystyle x_{3}} . The lowercase letters

16800-576: The majority of the unreported 9.5 million deaths believed to be direct deaths due the virus, rather than indirect deaths. Some deaths were because people with other conditions could not access medical services . A December 2022 WHO study estimated excess deaths from the pandemic during 2020 and 2021, again concluding ≈14.8 million excess early deaths occurred, reaffirming and detailing their prior calculations from May as well as updating them, addressing criticisms. These numbers do not include measures like years of potential life lost and may make

16950-417: The manipulation of statements within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations such as addition and multiplication . Elementary algebra is the main form of algebra taught in school and examines mathematical statements using variables for unspecified values. It seeks to determine for which values

17100-647: The matrices A = [ 9 3 − 13 2.3 0 7 − 5 − 17 0 ] , X = [ x 1 x 2 x 3 ] , B = [ 0 9 − 3 ] . {\displaystyle A={\begin{bmatrix}9&3&-13\\2.3&0&7\\-5&-17&0\end{bmatrix}},\quad X={\begin{bmatrix}x_{1}\\x_{2}\\x_{3}\end{bmatrix}},\quad B={\begin{bmatrix}0\\9\\-3\end{bmatrix}}.} Under some conditions on

17250-475: The method of completing the square . Many of these insights found their way to the ancient Greeks. Starting in the 6th century BCE, their main interest was geometry rather than algebra, but they employed algebraic methods to solve geometric problems. For example, they studied geometric figures while taking their lengths and areas as unknown quantities to be determined, as exemplified in Pythagoras ' formulation of

17400-399: The number 3 to the result. Other examples of algebraic expressions are 32 x y z {\displaystyle 32xyz} and 64 x 1 2 + 7 x 2 − c {\displaystyle 64x_{1}^{2}+7x_{2}-c} . Some algebraic expressions take the form of statements that relate two expressions to one another. An equation is

17550-470: The number of operations they use and the laws they obey. In mathematics education , abstract algebra refers to an advanced undergraduate course that mathematics majors take after completing courses in linear algebra. On a formal level, an algebraic structure is a set of mathematical objects, called the underlying set, together with one or several operations. Abstract algebra is primarily interested in binary operations , which take any two objects from

17700-511: The number of rows and columns, matrices can be added , multiplied , and sometimes inverted . All methods for solving linear systems may be expressed as matrix manipulations using these operations. For example, solving the above system consists of computing an inverted matrix A − 1 {\displaystyle A^{-1}} such that A − 1 A = I , {\displaystyle A^{-1}A=I,} where I {\displaystyle I}

17850-436: The numbers with variables, it is possible to express a general law that applies to any possible combination of numbers, like the commutative property of multiplication , which is expressed in the equation a × b = b × a {\displaystyle a\times b=b\times a} . Algebraic expressions are formed by using arithmetic operations to combine variables and numbers. By convention,

18000-495: The official names COVID-19 and SARS-CoV-2 on 11 February 2020. Tedros Adhanom Ghebreyesus explained: CO   for corona , VI   for virus , D   for disease and 19 for when the outbreak was first identified (31 December 2019). WHO additionally uses "the COVID-19 virus" and "the virus responsible for COVID-19" in public communications. WHO named variants of concern and variants of interest using Greek letters . The initial practice of naming them according to where

18150-425: The operation of addition. The neutral element is 0 and the inverse element of any number a {\displaystyle a} is − a {\displaystyle -a} . The natural numbers with addition, by contrast, do not form a group since they contain only positive integers and therefore lack inverse elements. Group theory examines the nature of groups, with basic theorems such as

18300-432: The operations are not restricted to regular arithmetic operations. For instance, the underlying set of the symmetry group of a geometric object is made up of geometric transformations , such as rotations , under which the object remains unchanged . Its binary operation is function composition , which takes two transformations as input and has the transformation resulting from applying the first transformation followed by

18450-427: The outbreak as having become a pandemic on 11 March. COVID-19 symptoms range from asymptomatic to deadly, but most commonly include fever, sore throat , nocturnal cough , and fatigue. Transmission of the virus is often through airborne particles . Mutations have produced many strains (variants) with varying degrees of infectivity and virulence . COVID-19 vaccines were developed rapidly and deployed to

18600-407: The pandemic 2021's leading cause of death . The time between symptom onset and death ranges from   6 to 41 days, typically about 14 days. Mortality rates increase as a function of age. People at the greatest mortality risk are the elderly and those with underlying conditions. The infection fatality ratio (IFR) is the cumulative number of deaths attributed to the disease divided by

18750-491: The pandemic evolved. Misinformation circulated through social media and mass media , and political tensions intensified . The pandemic raised issues of racial and geographic discrimination , health equity , and the balance between public health imperatives and individual rights . The WHO ended the PHEIC for COVID-19 on 5 May 2023. The disease has continued to circulate, but as of 2024, experts were uncertain as to whether it

18900-471: The population's age structure and the age-specific pattern of infections. There have also been reviews that have compared the fatality rate of this pandemic with prior pandemics, such as MERS-CoV. For comparison the infection mortality rate of seasonal flu in the United States is 0.1%, which is 13 times lower than COVID-19. Another metric in assessing death rate is the case fatality ratio (CFR), which

19050-449: The relation between field theory and group theory, relying on the fundamental theorem of Galois theory . Besides groups, rings, and fields, there are many other algebraic structures studied by algebra. They include magmas , semigroups , monoids , abelian groups , commutative rings , modules , lattices , vector spaces , algebras over a field , and associative and non-associative algebras . They differ from each other in regard to

19200-458: The risk of dying from seasonal influenza . In December 2020, a systematic review and meta-analysis estimated that population-weighted IFR was 0.5% to 1% in some countries (France, Netherlands, New Zealand, and Portugal), 1% to 2% in other countries (Australia, England, Lithuania, and Spain), and about 2.5% in Italy. This study reported that most of the differences reflected corresponding differences in

19350-430: The same axioms. The only difference is that the underlying set of the subalgebra is a subset of the underlying set of the algebraic structure. All operations in the subalgebra are required to be closed in its underlying set, meaning that they only produce elements that belong to this set. For example, the set of even integers together with addition is a subalgebra of the full set of integers together with addition. This

19500-496: The same infection may have different symptoms, and their symptoms may change over time. Three common clusters of symptoms have been identified: one respiratory symptom cluster with cough, sputum , shortness of breath, and fever; a musculoskeletal symptom cluster with muscle and joint pain, headache, and fatigue; a cluster of digestive symptoms with abdominal pain, vomiting, and diarrhoea. In people without prior ear, nose, and throat disorders, loss of taste combined with loss of smell

19650-543: The same operations while allowing variables in addition to regular numbers. Variables are symbols for unspecified or unknown quantities. They make it possible to state relationships for which one does not know the exact values and to express general laws that are true, independent of which numbers are used. For example, the equation 2 × 3 = 3 × 2 {\displaystyle 2\times 3=3\times 2} belongs to arithmetic and expresses an equality only for these specific numbers. By replacing

19800-401: The second algebraic structure plays the same role as the operation ∘ {\displaystyle \circ } does in the first algebraic structure. Isomorphisms are a special type of homomorphism that indicates a high degree of similarity between two algebraic structures. An isomorphism is a bijective homomorphism, meaning that it establishes a one-to-one relationship between

19950-442: The second as its output. Abstract algebra classifies algebraic structures based on the laws or axioms that its operations obey and the number of operations it uses. One of the most basic types is a group, which has one operation and requires that this operation is associative and has an identity element and inverse elements . An operation is associative if the order of several applications does not matter, i.e., if (

20100-459: The statements are true. To do so, it uses different methods of transforming equations to isolate variables. Linear algebra is a closely related field that investigates linear equations and combinations of them called systems of linear equations . It provides methods to find the values that solve all equations in the system at the same time, and to study the set of these solutions. Abstract algebra studies algebraic structures, which consist of

20250-444: The structures and patterns that underlie logical reasoning , exploring both the relevant mathematical structures themselves and their application to concrete problems of logic. It includes the study of Boolean algebra to describe propositional logic as well as the formulation and analysis of algebraic structures corresponding to more complex systems of logic . COVID-19 pandemic The COVID-19 pandemic (also known as

20400-410: The study of diverse types of algebraic operations and structures together with their underlying axioms , the laws they follow. Elementary algebra, also called school algebra, college algebra, and classical algebra, is the oldest and most basic form of algebra. It is a generalization of arithmetic that relies on variables and examines how mathematical statements may be transformed. Arithmetic

20550-485: The study of polynomials associated with elementary algebra towards a more general inquiry into algebraic structures, marking the emergence of abstract algebra . This approach explored the axiomatic basis of arbitrary algebraic operations. The invention of new algebraic systems based on different operations and elements accompanied this development, such as Boolean algebra , vector algebra , and matrix algebra . Influential early developments in abstract algebra were made by

20700-558: The sum of the Team and Power Rounds, then the Relay Round total. At the end, there is a set of tiebreaker questions to determine the top 20 participants, who receive cash prizes from D. E. Shaw & Co. Each student tied for the highest score (or the highest two scores if there are fewer than 20 tied with the top score) is given up to three tiebreaker questions, one at a time, with the goal of answering correctly as quickly as possible. As soon as

20850-528: The super relay round. Also in recent years, a song contest has become an informal event at ARML. Each school is allowed to have any number of their students perform a song related to mathematics, usually a parody of a popular song, with its lyrics replaced. The format of the ARML competition is based on the NYSML competition , but is generally considered more difficult than the NYSML competition. This format also inspired

21000-406: The theories of matrices and finite-dimensional vector spaces are essentially the same. In particular, vector spaces provide a third way for expressing and manipulating systems of linear equations. From this perspective, a matrix is a representation of a linear map: if one chooses a particular basis to describe the vectors being transformed, then the entries in the matrix give the results of applying

21150-411: The time from symptom onset to death has ranged from two to eight weeks. Prolonged prothrombin time and elevated C-reactive protein levels on admission to the hospital are associated with severe course of COVID-19 and with a transfer to intensive care units (ICU). Between 5% and 50% of COVID-19 patients experience long COVID , a condition characterised by long-term consequences persisting after

21300-413: The types of objects they describe and the requirements that their operations fulfill. Many are related to each other in that a basic structure can be turned into a more advanced structure by adding additional requirements. For example, a magma becomes a semigroup if its operation is associative. Homomorphisms are tools to examine structural features by comparing two algebraic structures. A homomorphism

21450-694: The typical convalescence period of the disease. The most commonly reported clinical presentations are fatigue and memory problems, as well as malaise , headaches, shortness of breath , loss of smell, muscle weakness , low fever and cognitive dysfunction . Many countries attempted to slow or stop the spread of COVID-19 by recommending, mandating or prohibiting behaviour changes, while others relied primarily on providing information. Measures ranged from public advisories to stringent lockdowns. Outbreak control strategies are divided into elimination and mitigation. Experts differentiate between elimination strategies (known as " zero-COVID ") that aim to completely stop

21600-510: The underlying set as inputs and map them to another object from this set as output. For example, the algebraic structure ⟨ N , + ⟩ {\displaystyle \langle \mathbb {N} ,+\rangle } has the natural numbers ( N {\displaystyle \mathbb {N} } ) as the underlying set and addition ( + {\displaystyle +} ) as its binary operation. The underlying set can contain mathematical objects other than numbers and

21750-400: The underlying sets and considers operations with more than two inputs, such as ternary operations . It provides a framework for investigating what structural features different algebraic structures have in common. One of those structural features concerns the identities that are true in different algebraic structures. In this context, an identity is a universal equation or an equation that

21900-543: The variants were identified (e.g. Delta began as the " Indian variant") is no longer common. A more systematic naming scheme reflects the variant's PANGO lineage (e.g., Omicron 's lineage is B.1.1.529) and is used for other variants. SARS-CoV-2 is a virus closely related to bat coronaviruses , pangolin coronaviruses, and SARS-CoV . The first known outbreak (the 2019–2020 COVID-19 outbreak in mainland China ) started in Wuhan , Hubei, China, in December 2019. Many early cases were linked to people who had visited

22050-443: Was accidentally released from a laboratory have been proposed, as of 2021 these were not supported by evidence. Official "case" counts refer to the number of people who have been tested for COVID-19 and whose test has been confirmed positive according to official protocols whether or not they experienced symptomatic disease. Due to the effect of sampling bias , studies which obtain a more accurate number by extrapolating from

22200-457: Was still a pandemic. Pandemics and their ends are not well-defined, and whether or not one has ended differs according to the definition used. As of 30 November 2024, COVID-19 has caused 7,075,455 confirmed deaths. The COVID-19 pandemic ranks as the fifth- deadliest pandemic or epidemic in history . In epidemiology , a pandemic is defined as "an epidemic occurring over a very wide area, crossing international boundaries, and usually affecting

22350-502: Was the most widely used. According to a June 2022 study, COVID-19 vaccines prevented an additional 14.4 million to 19.8 million deaths in 185 countries and territories from 8 December 2020 to 8 December 2021. On 8 November 2022, the first recombinant protein-based COVID-19 vaccine (Novavax's booster Nuvaxovid ) was authorised for use in adults in the United Kingdom. It has subsequently received endorsement/authorisation from

22500-403: Was translated into Latin as Liber Algebrae et Almucabola . The word entered the English language in the 16th century from Italian , Spanish , and medieval Latin . Initially, its meaning was restricted to the theory of equations , that is, to the art of manipulating polynomial equations in view of solving them. This changed in the 19th century when the scope of algebra broadened to cover

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