135-395: Following is a list of quartermaster corps , military units, active and defunct, with logistics duties: Logistics Logistics is the part of supply chain management that deals with the efficient forward and reverse flow of goods, services, and related information from the point of origin to the point of consumption according to the needs of customers. Logistics management
270-563: 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
405-403: 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
540-746: 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
675-429: 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
810-433: 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 ∘
945-402: 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
1080-437: 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
1215-652: 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
1350-623: 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
1485-404: 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
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#17328379455891620-526: A "Director of Operations" or a "Logistics Officer" working on similar problems. Furthermore, the term " supply chain management " originally referred to, among other issues, having an integrated vision of both production and logistics from point of origin to point of production. All these terms may suffer from semantic change as a side effect of advertising. Logistical activities can be divided into three main areas: order processing, inventory management, and freight transportation. Traditionally, order processing
1755-453: A call for professionals called supply chain logisticians. In business, logistics may have either an internal focus (inbound logistics) or an external focus (outbound logistics), covering the flow and storage of materials from point of origin to point of consumption, a key factor in supply-chain management . The main functions of a qualified logistician include inventory management , purchasing , transportation, warehousing , consultation, and
1890-404: 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
2025-461: 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
2160-442: A large number of goods to a user. Generally, there are three types of intermediaries, namely: agent/broker, wholesaler, and retailer. The nodes of a distribution network include: A logistic family is a set of products that share a common characteristic: weight and volumetric characteristics, physical storing needs (temperature, radiation, etc.), handling needs, order frequency, package size, etc. The following metrics may be used by
2295-467: 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
2430-412: 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 +
2565-547: A relatively consistent consumption rate regardless of war or peace. Some classes of supply have a linear demand relationship: as more troops are added, more supply items are needed; or as more equipment is used, more fuel and ammunition are consumed. Other classes of supply must consider a third variable besides usage and quantity: time. As equipment ages, more and more repair parts are needed over time, even when usage and quantity stay consistent. By recording and analyzing these trends over time and applying them to future scenarios,
2700-428: 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
2835-494: A significant need for logistics solutions and so have developed advanced implementations. Integrated logistics support (ILS) is a discipline used in military industries to ensure an easily supportable system with a robust customer service (logistic) concept at the lowest cost and in line with (often high) reliability, availability, maintainability, and other requirements, as defined for the project. In military logistics , Logistics Officers manage how and when to move resources to
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#17328379455892970-447: 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}
3105-503: A single customer's demand to be fulfilled efficiently. Track and tracing , which is an essential part of production logistics due to product safety and reliability issues, is also gaining importance, especially in the automotive and medical industries. Construction logistics has been employed by civilizations for thousands of years as the various human civilizations tried to build the best possible works of construction for living and protection. Now, construction logistics has emerged as
3240-484: A specialist provider. The term production logistics describes logistic processes within a value-adding system (ex, a factory or a mine). Production logistics aims to ensure that each machine and workstation receives the right product in the correct quantity and quality at the right time. The concern is with production, testing, transportation, storage, and supply. Production logistics can operate in existing as well as new plants. Since manufacturing in an existing plant
3375-435: 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
3510-736: A stockpile of finished goods beforehand can reduce the frequency of transportation to and from the customers and cope with the randomness of customer demands. However, maintaining an inventory requires capital investment in finished goods and maintaining a warehouse. Storage and order picking occupy most of the warehouse maintenance cost. Freight transportation forms a vital part of logistics and allows access to broad markets as goods can be transported to hundreds or thousands of kilometers away. Freight transportation accounts for two-thirds of logistical costs and significantly impacts customer service. Transportation policies and warehouse management are closely intertwined. The rise of commercial transactions through
3645-527: 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
3780-421: A vital part of construction. In the past few years, construction logistics has emerged as a different field of knowledge and study within supply chain management and logistics. The Seven R's is a popular concept used to enforce best practices in logistics management which consists of the following: In military science, maintaining one's supply lines while disrupting those of the enemy is a crucial—some would say
3915-710: A warehouse. A distribution network would require several intermediaries to bring consumer or industrial goods from manufacturers to a user. Intermediaries would markup the costs of the products during distribution, but benefit users by providing lower transportation costs than the manufacturers. The number of intermediaries required for the distribution network depends upon the types of goods being distributed. For example, consumer goods such as cosmetics and handicrafts may not require any intermediaries as they can be sold door-to-door or can be obtained from local flea markets. For industrial goods such as raw materials and equipment, intermediaries are not needed because manufacturers can sell
4050-400: 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
4185-397: 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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4320-438: 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
4455-685: Is a component that holds the supply chain together. The resources managed in logistics may include tangible goods such as materials, equipment, and supplies, as well as food and other consumable items. In military logistics , it is concerned with maintaining army supply lines with food, armaments, ammunition, and spare parts apart from the transportation of troops themselves. Meanwhile, civil logistics deals with acquiring, moving, and storing raw materials, semi-finished goods, and finished goods. For organisations that provide garbage collection, mail deliveries, public utilities, and after-sales services, logistical problems must be addressed. Logistics deals with
4590-416: Is a constantly changing process, machines are exchanged and new ones added, which allows for improving the production logistics system accordingly. Production logistics provides the means to achieve customer response and capital efficiency. Production logistics becomes more important with decreasing batch sizes. In many industries (e.g. mobile phones ), the short-term goal is a batch size of one, allowing even
4725-475: 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
4860-504: 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
4995-415: 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
5130-487: 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
5265-941: 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
5400-414: 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
5535-629: 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
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5670-540: Is measured in the Logistics Performance Index . Distribution logistics has, as its main task, the delivery of the finished products to the customer. It consists of order processing, warehousing, and transportation. Distribution logistics is necessary because production time, place, and quantity differ with the time, place, and quantity of consumption. Disposal logistics has the main function of reducing logistics cost(s) and enhancing service(s) related to
5805-408: 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
5940-482: 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
6075-635: Is presumably the origin of the term logistic in logistic growth and related terms. Some sources give this instead as the source of logistics , either ignorant of Jomini's statement that it was derived from logis , or dubious and instead believing it was in fact of Greek origin, or influenced by the existing term of Greek origin. Jomini originally defined logistics as: ... l'art de bien ordonner les marches d'une armée, de bien combiner l'ordre des troupes dans les colonnes, les tems [temps] de leur départ, leur itinéraire, les moyens de communications nécessaires pour assurer leur arrivée à point nommé ... ...
6210-400: 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
6345-403: 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
6480-414: 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
6615-425: 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
6750-421: 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,
6885-472: 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
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#17328379455897020-718: 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
7155-466: 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
7290-446: 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,
7425-548: The New Oxford American Dictionary defines logistics as "the detailed coordination of a complex operation involving many people, facilities, or supplies", and the Oxford Dictionary on-line defines it as "the detailed organization and implementation of a complex operation". As such, logistics is commonly seen as a branch of engineering that creates "people systems" rather than "machine systems". According to
7560-538: The European Union , logistics costs were 8.8% to 11.5% of GDP as of 1993. Dedicated simulation software can model, analyze, visualize, and optimize logistics' complexity. Minimizing resource use is a common motivation in all logistics fields. A professional working in logistics management is called a logistician. The term logistics is attested in English from 1846, and is from French: logistique , where it
7695-826: The Romans during the Punic Wars and the success of the Anglo-Portuguese army in the Peninsula War was due to the effectiveness of Wellington's supply system, despite the numerical disadvantage. The defeat of the British in the American War of Independence and the defeat of the Axis in the African theater of World War II are attributed by some scholars to logistical failures. Militaries have
7830-466: The US Armed Forces can accurately supply troops with the items necessary at the precise moment they are needed. History has shown that good logistical planning creates a lean and efficient fighting force. The lack thereof can lead to a clunky, slow, and ill-equipped force with too much or too little supply. One definition of business logistics speaks of "having the right item in the right quantity at
7965-414: 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
8100-547: 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
8235-525: 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,
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#17328379455898370-602: 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
8505-461: 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}
8640-653: 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,
8775-528: 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
8910-464: 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
9045-402: 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
9180-636: The Council of Supply Chain Management Professionals (previously the Council of Logistics Management), logistics is the process of planning, implementing and controlling procedures for the efficient and effective transportation and storage of goods including services and related information from the point of origin to the point of consumption for the purpose of conforming to customer requirements and includes inbound, outbound, internal and external movements. Academics and practitioners traditionally refer to
9315-481: 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
9450-548: 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
9585-592: 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
9720-561: 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
9855-454: 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
9990-401: 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
10125-421: The art of well-ordering the functionings of an army, of well combining the order of troops in columns, the times of their departure, their itinerary, the means of communication necessary to assure their arrival at a named point ... The Oxford English Dictionary defines logistics as "the branch of military science relating to procuring, maintaining and transporting material, personnel and facilities". However,
10260-443: 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
10395-459: The company to organize its products in different families: Other metrics may present themselves in both physical or monetary form, such as the standard inventory turnover . Unit loads are combinations of individual items which are moved by handling systems, usually employing a pallet of normed dimensions. Algebra Algebra is the branch of mathematics that studies certain abstract systems , known as algebraic structures , and
10530-514: The company's autonomy, or minimizing procurement costs while maximizing security within the supply process. Advance logistics consists of the activities required to set up or establish a plan for logistics activities to occur. Global logistics is technically the process of managing the "flow" of goods through a supply chain from its place of production to other parts of the world. This often requires an intermodal transport system via ocean, air, rail, and truck. The effectiveness of global logistics
10665-416: 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
10800-593: 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
10935-455: 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
11070-466: The disposal of waste produced during a business's operation. Reverse logistics denotes all those reusing products and materials operations. The reverse logistics process includes the management and the sale of surpluses, as well as products being returned to vendors from buyers. It is "the process of planning, implementing, and controlling the efficient, cost-effective flow of raw materials, in-process inventory, finished goods, and related information from
11205-469: 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
11340-496: 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
11475-403: 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
11610-404: 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
11745-507: The event of an emergency. The reason for enlisting emergency logistics services could be a production delay or anticipated production delay, or an urgent need for specialized equipment to prevent events such as aircraft being grounded (also known as " aircraft on ground "—AOG), ships being delayed, or telecommunications failure. Humanitarian logistics involves governments, the military, aid agencies , donors, non-governmental organizations, and emergency logistics services are typically sourced from
11880-612: 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
12015-429: 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
12150-493: The forward and reverse flows. This can be achieved through intermodal freight transport , path optimization, vehicle saturation, and city logistics . RAM logistics (see also Logistic engineering ) combines both business logistics and military logistics since it concerns highly complicated technological systems for which reliability , availability and maintainability are essential, e.g., weapon system and military supercomputers. Asset control logistics : companies in
12285-438: 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
12420-413: 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
12555-739: The inbound movement of materials, parts, or unfinished inventory from suppliers to manufacturing or assembly plants, warehouses, or retail stores. Outbound logistics is the process related to the storage and movement of the final product. The related information flows from the end of the production line to the end user. Given the services performed by logisticians, the main fields of logistics can be broken down as follows: Procurement logistics consists of market research , requirements planning, make-or-buy decisions, supplier management, ordering, and order control. The targets in procurement logistics might be contradictory: maximizing efficiency by concentrating on core competencies, outsourcing while maintaining
12690-444: The internet gives rise to the need for "e-logistics". Compared to traditional logistics, e-logistics handles parcels valued at less than a hundred US dollars to customers scattered at various destinations worldwide. In e-logistics, customers' demands come in waves when compared to traditional logistics, where the demand is consistent. Inbound logistics is one of the primary logistics processes concentrating on purchasing and arranging
12825-495: 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
12960-607: 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
13095-620: 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
13230-426: 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
13365-472: 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
13500-417: 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
13635-647: 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
13770-475: 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
13905-513: The most crucial—element of military strategy , since an armed force without resources and transportation is defenseless. The historical leaders Hannibal , Alexander the Great , and the Duke of Wellington are considered to have been logistical geniuses: Alexander's expedition benefited considerably from his meticulous attention to the provisioning of his army, Hannibal is credited to have "taught logistics" to
14040-565: The movements of materials or products from one facility to another; it does not include material flow within the production or assembly plants, such as production planning or single-machine scheduling . Logistics occupies a significant amount of the operational cost of an organisation or country. Logistical costs of organizations in the United States incurred about 11% of the United States national gross domestic product (GDP) as of 1997. In
14175-399: 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
14310-470: 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
14445-511: 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}
14580-436: 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,
14715-702: The officers of the general staff were named: marshall of lodgings, major-general of lodgings; from there came the term of logistics [ logistique ], which we employ to designate those who are in charge of the functioning of an army. The term is credited to Jomini, and the term and its etymology criticized by Georges de Chambray in 1832, writing: Logistique : Ce mot me paraît être tout-à-fait nouveau, car je ne l'avais encore vu nulle part dans la littérature militaire. … il paraît le faire dériver du mot logis , étymologie singulière … Logistic : This word appears to me to be completely new, as I have not yet seen it anywhere in military literature. … he appears to derive it from
14850-425: 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
14985-432: 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
15120-426: The organizing and planning of these activities. Logisticians combine professional knowledge of each of these functions to coordinate resources in an organization. There are two fundamentally different forms of logistics: one optimizes a steady flow of material through a network of transport links and storage nodes, while the other coordinates a sequence of resources to carry out some project , such as restructuring
15255-585: The places they are needed. Supply chain management in military logistics often deals with a number of variables in predicting cost, deterioration, consumption , and future demand. The United States Armed Forces ' categorical supply classification was developed in such a way that categories of supply with similar consumption variables are grouped together for planning purposes. For instance, peacetime consumption of ammunition and fuel will be considerably lower than wartime consumption of these items, whereas other classes of supply such as subsistence and clothing have
15390-409: The point of consumption to the point of origin to recapture value or proper disposal." More precisely, reverse logistics moves goods from their typical final destination to capture value or proper disposal. The opposite of reverse logistics is forward logistics . ' Green logistics describes all attempts to measure and minimize the ecological impact of logistics activities, including all activities of
15525-429: The products made by a factory are ready for consumption they still need to be moved along the distribution network according to some logic, and the distribution center aggregates and processes orders coming from different areas of the territory. That being said, from a modeling perspective, there are similarities between operations management and logistics, and companies sometimes use hybrid professionals, with for example
15660-449: 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
15795-476: The retail channels, both organized retailers and suppliers, often deploy assets required for the display, preservation, and promotion of their products. Some examples are refrigerators, stands, display monitors, seasonal equipment, poster stands & frames. Emergency logistics (or humanitarian logistics ) is a term used by the logistics, supply chain, and manufacturing industries to denote specific time-critical modes of transport used to move goods rapidly in
15930-473: The right time at the right place for the right price in the right condition to the right customer". Business logistics incorporates all industry sectors and aims to manage the fruition of project life cycles , supply chains , and resultant efficiencies. The term business logistics has evolved since the 1960s due to the increasing complexity of supplying businesses with materials and shipping out products in an increasingly globalized supply chain, leading to
16065-430: 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
16200-543: 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
16335-401: 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
16470-442: 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 (
16605-459: 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
16740-410: 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
16875-485: 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
17010-415: The terms French : maréchal des logis , lit. 'marshall of lodgings' and French : major-général des logis , lit. 'major-general of lodging': Autrefois les officiers de l’état-major se nommaient: maréchal des logis, major-général des logis; de là est venu le terme de logistique, qu’on emploie pour désigner ce qui se rapporte aux marches d’une armée. Formerly
17145-408: The terms operations or production management when referring to physical transformations taking place in a single business location (factory, restaurant or even bank clerking) and reserve the term logistics for activities related to distribution, that is, moving products on the territory. Managing a distribution center is seen, therefore, as pertaining to the realm of logistics since, while in theory,
17280-406: 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
17415-413: 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
17550-510: 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
17685-400: 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
17820-473: The word lodgings [ logis ], a peculiar etymology … Chambray also notes that the term logistique was present in the Dictionnaire de l'Académie française as a synonym for algebra . The French word: logistique is a homonym of the existing mathematical term, from Ancient Greek : λογῐστῐκός , romanized : logistikós , a traditional division of Greek mathematics ; the mathematical term
17955-411: Was a time-consuming activity that could take up to 70% of the order-cycle time. However, with new technologies such as bar code scanning, computers, and network connection, customer orders can quickly reach the seller in no time, and the availability of stocks can be checked in real time. The purpose of having an inventory is to reduce the overall logistical cost while improving service to customers. Having
18090-463: Was either coined or popularized by Swiss military officer and writer Antoine-Henri Jomini , who defined it in his Summary of the Art of War ( Précis de l'Art de la Guerre ). The term appears in the 1830 edition, then titled Analytic Table ( Tableau Analytique ), and Jomini explains that it is derived from French : logis , lit. 'lodgings' (cognate to English lodge ), in
18225-403: 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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