Algebraic structure

In mathematics, more in abstract algebra and universal algebra, an algebraic structure consists of a set A, a collection of operations on A of finite arity, a finite set of identities, known as axioms, that these operations must satisfy. Some algebraic structures involve another set. In the context of universal algebra, the set A with this structure is called an algebra, while, in other contexts, it is called an algebraic structure, the term algebra being reserved for specific algebraic structures that are vector spaces over a field or modules over a commutative ring. Examples of algebraic structures with a single underlying set include groups, rings and lattices. Examples of algebraic structures with two underlying sets include vector spaces and algebras; the properties of specific algebraic structures are studied in abstract algebra. The general theory of algebraic structures has been formalized in universal algebra; the language of category theory is used to express and study relationships between different classes of algebraic and non-algebraic objects.

This is because it is sometimes possible to find strong connections between some classes of objects, sometimes of different kinds. For example, Galois theory establishes a connection between certain fields and groups: two algebraic structures of different kinds. Addition and multiplication on numbers are the prototypical example of an operation that combines two elements of a set to produce a third; these operations obey several algebraic laws. For example, a + = + examples of the associative law. A + b = b + a, ab = ba, the commutative law. Many systems studied by mathematicians have operations that obey some, but not all, of the laws of ordinary arithmetic. For example, rotations of objects in three-dimensional space can be combined by performing the first rotation and applying the second rotation to the object in its new orientation; this operation on rotations can fail the commutative law. Mathematicians give names to sets with one or more operations that obey a particular collection of laws, study them in the abstract as algebraic structures.

When a new problem can be shown to follow the laws of one of these algebraic structures, all the work, done on that category in the past can be applied to the new problem. In full generality, algebraic structures may involve an arbitrary number of sets and operations that can combine more than two elements, but this article focuses on binary operations on one or two sets; the examples here are by no means a complete list, but they are meant to be a representative list and include the most common structures. Longer lists of algebraic structures may be found in the external links and within Category:Algebraic structures. Structures are listed in approximate order of increasing complexity. Simple structures: no binary operation: Set: a degenerate algebraic structure S having no operations. Pointed set: S has one or more distinguished elements 0, 1, or both. Unary system: S and a single unary operation over S. Pointed unary system: a unary system with S a pointed set. Group-like structures: one binary operation.

The binary operation can be indicated by any symbol, or with no symbol as is done for ordinary multiplication of real numbers. Magma or groupoid: S and a single binary operation over S. Semigroup: an associative magma. Monoid: a semigroup with identity element. Group: a monoid with a unary operation, giving rise to inverse elements. Abelian group: a group whose binary operation is commutative. Semilattice: a semigroup whose operation is idempotent and commutative; the binary operation can be called either join. Quasigroup: a magma obeying the latin square property. A quasigroup may be represented using three binary operations. Loop: a quasigroup with identity. Ring-like structures or Ringoids: two binary operations called addition and multiplication, with multiplication distributing over addition. Semiring: a ringoid such that S is a monoid under each operation. Addition is assumed to be commutative and associative, the monoid product is assumed to distribute over the addition on both sides, the additive identity 0 is an absorbing element in the sense that 0 x = 0 for all x.

Near-ring: a semiring whose additive monoid is a group. Ring: a semiring whose additive monoid is an abelian group. Lie ring: a ringoid whose additive monoid is an abelian group, but whose multiplicative operation satisfies the Jacobi identity rather than associativity. Commutative ring: a ring in which the multiplication operation is commutative. Boolean ring: a commutative ring with idempotent multiplication operation. Field: a commutative ring which contains a multiplicative inverse for every nonzero element. Kleene algebras: a semiring with idempotent addition and a unary operation, the Kleene star, satisfying additional properties. *-algebra: a ring with an additional unary operation satisfying additional properties. Lattice structures: two or more binary operations, including operations called meet and join, connected by the absorption law. Complete lattice: a lattice in which arbitrary meet and joins exist. Bounded lattice: a lattice with a greatest element and least element. Complemented lattice: a bounded lattice with a unary operation, denoted by postfix ⊥.

The join of an element with its complement is the greatest element, the meet of the two elements is the least element. Modular lattice: a lattice whose elements satisfy the additional modular identity. Distributive lattice: a lattice in which

Lost-foam casting

Lost-foam casting is a type of evaporative-pattern casting process, similar to investment casting except foam is used for the pattern instead of wax. This process takes advantage of the low boiling point of polymer foams to simplify the investment casting process by removing the need to melt the wax out of the mold. First, a pattern is made from polystyrene foam. For small volume runs the pattern can be hand machined from a solid block of foam. If the volume is large the pattern can be mass-produced by a process similar to injection molding. Pre-expanded beads of polystyrene are injected into a preheated aluminum mold at low pressure. Steam is applied to the polystyrene which causes it to expand more to fill the die; the final pattern is 97.5% air and 2.5% polystyrene. Pre-made pouring basins and risers can be hot glued to the pattern to finish it. Next, the foam cluster is coated with ceramic investment known as the refractory coating, via dipping, spraying or flow coating. After the coating dries, the cluster is placed into a flask and backed up with un-bonded sand, compacted using a vibration table.

The refractory coating captures all of the detail in the foam model and creates a barrier between the smooth foam surface and the coarse sand surface. Secondly it controls permeability, which allows the gas created by the vaporized foam pattern to escape through the coating and into the sand. Controlling permeability is a crucial step to avoid sand erosion, it forms a barrier so that molten metal does not penetrate or cause sand erosion during pouring. Once the sand is compacted, the mold is ready to be poured. Automatic pouring is used in LFC, as the pouring process is more critical than in conventional foundry practice. There is no bake-out phase, as for lost-wax; the melt is poured directly into the foam-filled mold. As the foam is of low density, the waste gas produced by this is small and can escape through mold permeability, as for the usual outgassing control. Cast metals include cast irons, aluminium alloys and nickel alloys; the size range is from 0.5 kg to several tonnes. The minimum wall thickness is 2.5 mm and there is no upper limit.

Typical surface finishes are from 2.5 to 25 µm RMS. Typical linear tolerances are ±0.005 mm/mm. This casting process is advantageous for complex castings that would require cores, it is dimensionally accurate, maintains an excellent surface finish, requires no draft, has no parting lines so no flash is formed. The un-bonded sand of lost foam casting can be much simpler to maintain than green sand and resin bonded sand systems. Lost foam is more economical than investment casting because it involves fewer steps. Risers are not required due to the nature of the process. Foam is easy to manipulate and glue, due to its unique properties; the flexibility of LFC allows for consolidating the parts into one integral component. The two main disadvantages are that pattern costs can be high for low volume applications and the patterns are damaged or distorted due to their low strength. If a die is used to create the patterns there is a large initial cost. Lost-foam casting was invented in the early fifties by Canadian sculptor Armand Vaillancourt.

Public recognition of the benefits of LFC was made by General Motors in the mid 1980s when it announced its new car line, would utilize LFC for production of all engine blocks, cylinder heads, differential carriers, transmission cases. Full-mold casting Degarmo, E. Paul. Materials and Processes in Manufacturing, Wiley, ISBN 0-471-65653-4. Kalpakjian, Serope.

Partie de campagne

Partie de campagne is a 1936 French featurette written and directed by Jean Renoir. It was released as A Day in the Country in the United States; the film is based on the short story "Une partie de campagne" by Guy de Maupassant, a friend of Renoir's father, the renowned painter Auguste Renoir. It chronicles a love affair over a single summer afternoon in 1860 along the banks of the Seine. Renoir never finished filming due to weather problems, but producer Pierre Braunberger turned the material into a release in 1946, ten years after it was shot. Joseph Burstyn released the film in the U. S. in 1950. Monsieur Dufour, a shop-owner from Paris, takes his family for a day of relaxation in the country; when they stop for lunch at the roadside restaurant of Poulain, two young men there and Rodolphe, take an interest in Dufour's daughter Henriette and wife Madame Dufour. They scheme to get the two women off alone with them, they offer to row them along the river in their skiffs, while they divert Dufour and his shop assistant and future son-in-law, Anatole, by lending them some fishing poles.

Though Rodolphe had arranged beforehand to take Henriette, Henri maneuvers it so that she gets into his skiff. Rodolphe good-naturedly settles for Madame Dufour. Henri rows to a secluded spot on the riverbank which he refers to as his "private office". Though Henriette resists his amorous advances, she gives in, he asks her to come see him again, but she says that her father would never permit her to venture into the countryside by herself. Years pass, Henriette marries Anatole. One day, they end up at the place. While Anatole dozes, his wife takes a walk, encounters Henri. With tears in her eyes, she reminisces about their brief time together; when Anatole wakes up, Henri hides until they leave. Sylvia Bataille as Henriette Georges D'Arnoux as Henri Jane Marken as Madame Dufour André Gabriello as Monsieur Dufour Jacques B. Brunius as Rodolphe Paul Temps as Anatole Gabrielle Fontan as Grandmother Jean Renoir as Uncle Poulain Marguerite Renoir as Waitress Future leading directors Jacques Becker and Luchino Visconti worked as Renoir's assistant directors.

The film was shot in July, soon after France had elected the Popular Front government, employers had negotiated the Matignon agreement, providing wage increases, 40-hour weeks, trade union rights, paid holidays and improved social services. A Day in the Country on IMDb A Day in the Country at The Film Journal A Day in the Country: Jean Renoir’s Sunday Outing an essay by Gilberto Perez at the Criterion Collection