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The dual of Brianchon's theorem (Casey 1888, p. 146), discovered by B. Pascal in 1640 when he was just 16 years old (Leibniz 1640; Wells 1986, p. 69). It states that, given a (not necessarily regular, or even convex) hexagon inscribed in a conic section, the three pairs of the continuations of opposite sides meet on a straight line, called the Pascal line.In 1847, Möbius (1885) published the following generalization of Pascal's theorem: if all intersection points (except possibly one) of the lines prolonging two opposite sides of a -gon inscribed in a conic section are collinear, then the same is true for the remaining point.

A theorem, also called a unicity theorem, stating the uniqueness of a mathematical object, which usually means that there is only one object fulfilling given properties, or that all objects of a given class are equivalent (i.e., they can be represented by the same model). This is often expressed by saying that the object is uniquely determined by a certain set of data. The word unique is sometimes replaced by essentially unique, whenever one wants to stress that the uniqueness is only referred to the underlying structure, whereas the form may vary in all ways that do not affect the mathematical content.The object of many uniqueness theorems is the solution to a problem or an equation; in such cases, a uniqueness theorem is normally combined with an existence theorem.

A theorem stating the existence of an object, such as the solution to a problem or equation. Strictly speaking, it need not tell how many such objects there are, nor give hints on how to find them. Some existence theorems give explicit formulas for solutions (e.g., Cramer's rule), others describe in their proofs iteration processes for approaching them (e.g., Bolzano-Weierstrass theorem), while others are settled by nonconstructive proofs which simply deduce the necessity of solutions without indicating any method for determining them (e.g., the Brouwer fixed point theorem, which is proved by reductio ad absurdum, showing that the nonexistence would lead to a contradiction).

According to G. Pólya, the method of finding geometric objects by intersection. 1. For example, the centers of all circles tangent to a straight line at a given point lie on a line that passes through and is perpendicular to . 2. In addition, the circle centered at with radius is the locus of the centers of all circles of radius passing through . The intersection of and consists of two points and which are the centers of two circles of radius tangent to at .Many constructions with straightedge and compass are based on this method, as, for example, the construction of the center of a given circle by means of the perpendicular bisector theorem.

A description of an object by properties that are different from those mentioned in its definition, but are equivalent to them. The following list gives a number of examples.1. A rational number is defined as the quotient of two integers, but it can be characterized as a number admitting a finite or repeating decimal expansion. 2. An equilateral triangle is defined as a triangle having three equal sides, but it can be characterized as a triangle having two angles of . 3. A real square matrix is nonsingular, by definition, if it admits a matrix inverse, but it can be characterized by the condition that its determinant be nonzero. Of course, a characterization should not merely be a rephrasing of the definition, but should give an entirely new description, which is useful because it contains a simpler formulation, can be verified more easily, is interesting because it places the object in another context, or unveils unexpected links between different..

A finite simple group of Lie-type. The following table summarizes the types of twisted Chevalley groups and their respective orders. In the table, denotes a prime power and the superscript denotes the order of the twisting automorphism.grouporder

There are a number of meanings for the word "arc" in mathematics. In general, an arc is any smooth curve joining two points. The length of an arc is known as its arc length.In a graph, a graph arc isan ordered pair of adjacent vertices.In particular, an arc is any portion (other than the entire curve) of the circumference of a circle. An arc corresponding to the central angle is denoted . Similarly, the size of the central angle subtended by this arc (i.e., the measure of the arc) is sometimes (e.g., Rhoad et al. 1984, p. 421) but not always (e.g., Jurgensen 1963) denoted .The center of an arc is the center of the circle of whichthe arc is a part.An arc whose endpoints lie on a diameter of a circleis called a semicircle.For a circle of radius , the arc length subtended by a central angle is proportional to , and if is measured in radians, then the constant of proportionality is 1, i.e.,(1)The length of the chord connecting the arc's endpointsis(2)As..

The symbol , used for the union of sets, and, sometimes, also for the logical connective OR instead of the symbol (vee). In fact, for any two sets and and this equivalence demonstrates the connection between the set-theoretical and the logical meaning.

The theorem in set theory and logic that for all sets and ,(1)where denotes complement set of and is the empty set. The set is depicted in the above Venn diagram and clearly coincides with iff is empty.The corresponding theorem in a Boolean algebra states that for all elements of ,(2)The version of Poretsky's Law for logic can be derived from (2) using the rules of propositional calculus, namely for all propositions and ,(3)where "is equivalent to" means having the same truth table. In fact, in the following table, the values in the second and in the third column coincide if and only if the value in the first column is 0.( and not ) or (not and )000011101110

A statement claiming the existence of an object with given properties. In the language of set theory it can be formulated as follows,where is the universal set and is a given set contained in it. In other words, it states that set is nonempty.

A metatheorem in mathematical logic also known under the name "conditional proof." It states that if the sentential formula can be derived from the set of sentential formulas , then the sentential formula can be derived from .In a less formal setting, this means that if a thesis can be proven under the hypotheses , then one can prove that implies under hypothesis .

The Galton board, also known as a quincunx or bean machine, is a device for statistical experiments named after English scientist Sir Francis Galton. It consists of an upright board with evenly spaced nails (or pegs) driven into its upper half, where the nails are arranged in staggered order, and a lower half divided into a number of evenly-spaced rectangular slots. The front of the device is covered with a glass cover to allow viewing of both nails and slots. In the middle of the upper edge, there is a funnel into which balls can be poured, where the diameter of the balls must be much smaller than the distance between the nails. The funnel is located precisely above the central nail of the second row so that each ball, if perfectly centered, would fall vertically and directly onto the uppermost point of this nail's surface (Kozlov and Mitrofanova 2002). The figure above shows a variant of the board in which only the nails that can potentially be hit by a ball..

The second Mersenne prime , which is itself the exponent of Mersenne prime . It gives rise to the perfect number It is a Gaussian prime, but not an Eisenstein prime, since it factors as , where is a primitive cube root of unity. It is the smallest non-Sophie Germain prime. It is also the smallest non-Fermat prime, and as such is the smallest number of faces of a regular polygon (the heptagon) that is not constructible by straightedge and compass.It occurs as a sacred number in the Bible and in various other traditions. In Babylonian numerology it was considered as the perfect number, the only number between 2 and 10 which is not generated (divisible) by any other number, nor does it generate (divide) any other number.Words referring to number seven may have the prefix hepta-, derived from the Greek -) (heptic), or sept- (septuple), derived from the Latin septem...

The smallest composite squarefree number (), and the third triangular number (). It is the also smallest perfect number, since . The number 6 arises in combinatorics as the binomial coefficient , which appears in Pascal's triangle and counts the 2-subsets of a set with 4 elements. It is also equal to (3 factorial), the number of permutations of three objects, and the order of the symmetric group (which is the smallest non-Abelian group).Six is indicated by the Latin prefix sex-, as in sextic, or by the Greek prefix hexa- (-), as in hexagon, hexagram, or hexahedron.The six-fold symmetry is typical of crystals such as snowflakes. A mathematical and physical treatment can be found in Kepler (Halleux 1975), Descartes (1637), Weyl (1952), and Chandrasekharan (1986).

The third prime number, which is also the second Fermat prime, the third Sophie Germain prime, and Fibonacci number . It is an Eisenstein prime, but not a Gaussian prime, since it factors as . It is the hypotenuse of the smallest Pythagorean triple: 3, 4, 5. For the Pythagorean school, the number 5 was the number of marriage, since it is was the sum of the first female number (2) and the first male number (3). The magic symbol of the pentagram was also based on number 5; it is a star polygon with the smallest possible number of sides, and is formed by the diagonals of a regular pentagon. These intersect each other according to the golden ratio .There are five Platonic solids. In algebra, five arises in Abel's impossibility theorem as the smallest degree for which an algebraic equation with general coefficients is not solvable by radicals. According to Galois theory, this property is a consequence of the fact that 5 is the smallest positive integer such that..

The smallest positive composite number and the first even perfect square. Four is the smallest even number appearing in a Pythagorean triple: 3, 4, 5. In the numerology of the Pythagorean school, it was the number of justice. The sacred tetraktýs (10) was the sum of the first four numbers, depicted as a triangle with two equal sides of length 4.4 is the highest degree for which an algebraic equation is always solvable by radicals. It is the smallest order of a field which is not a prime field, and the smallest order for which there exist two nonisomorphic finite groups (finite group C2×C2 and the cyclic group C4). It is the smallest number of faces of a regular polyhedron, the tetrahedron. In the three-dimensional Euclidean space, there is exactly one sphere passing through four noncoplanar points. Four is the number of dimensions of space-time.Words related to number four are indicated by the Greek prefix tetra (e.g., tetromino) or by..

A set-theoretic term having a number of different meanings. Fraenkel (1953, p. 37) used the term as a synonym for "finite set." However, according to Russell's definition (Russell 1963, pp. 21-22), an inductive set is a nonempty partially ordered set in which every element has a successor. An example is the set of natural numbers , where 0 is the first element, and the others are produced by adding 1 successively.Roitman (1990, p. 40) considers the same construction in a more abstract form: the elements are sets, 0 is replaced by the empty set , and the successor of every element is the set . In particular, every inductive set contains a sequence of the formFor many other authors (e.g., Bourbaki 1970, pp. 20-21; Pinter 1971, p. 119), an inductive set is a partially ordered set in which every totally ordered subset has an upper bound, i.e., it is a set fulfilling the assumption of Zorn's lemma.The versions..

A set whose elements can be numbered through from 1 to , for some positive integer . The number is called the cardinal number of the set, and is often denoted or . In other words, is equipollent to the set . We simply say that has elements. The empty set is also considered as a finite set, and its cardinal number is 0.A finite set can also be characterized as a set which is not infinite, i.e., as a set which is not equipollent to any of its proper subsets. In fact, if , and , a certain number of elements of do not belong to , so that .For all , the number of subsets having exactly elements (the so-called k-subsets, or combinations of elements out of ) is equal to the binomial coefficient(1)(2)Hence the number of subsets of (i.e., the cardinal number of its power set) is(3)(4)(5)by virtue of the binomial theorem.Assigning to each k-subset of its complement set defines a one-to-one correspondence between the set of k-subsets and the set of -subsets of . This proves the..

A map is called bijective if it is both injective and surjective. A bijective map is also called a bijection. A function admits an inverse (i.e., " is invertible") iff it is bijective.Two sets and are called bijective if there is a bijective map from to . In this sense, "bijective" is a synonym for "equipollent" (or "equipotent"). Bijectivity is an equivalence relation on the class of sets.

A metatheorem stating that every theorem on partially ordered sets remains true if all inequalities are reversed. In this operation, supremum must be replaced by infimum, maximum with minimum, and conversely. In a lattice, this means that meet and join must be interchanged, and in a Boolean algebra, 1 and 0 must be switched.Each of de Morgan's two laws can be derived from the other by duality.

The distance between two vertices and of a finite graph is the minimum length of the paths connecting them (i.e., the length of a graph geodesic). If no such path exists (i.e., if the vertices lie in different connected components), then the distance is set equal to . In a grid graph the distance between two vertices is the sum of the "vertical" and the "horizontal" distances (right figure above).The matrix consisting of all distances from vertex to vertex is known as the all-pairs shortest path matrix, or more simply, the graph distance matrix.

Two points on a compact Riemann surface such that lies on every geodesic passing through , and conversely. An oriented surface where every point belongs to a Wiedersehen pair is called a Wiedersehen surface.The name is the German word for "seeing again" and was introduced by Blaschke.

A linear system of equations is a set of linear equations in variables (sometimes called "unknowns"). Linear systems can be represented in matrix form as the matrix equation(1)where is the matrix of coefficients, is the column vector of variables, and is the column vector of solutions.If , then the system is (in general) overdetermined and there is no solution.If and the matrix is nonsingular, then the system has a unique solution in the variables. In particular, as shown by Cramer's rule, there is a unique solution if has a matrix inverse . In this case,(2)If , then the solution is simply . If has no matrix inverse, then the solution set is the translate of a subspace of dimension less than or the empty set.If two equations are multiples of each other, solutions are ofthe form(3)for a real number. More generally, if , then the system is underdetermined. In this case, elementary row and column operations can be used to solve the system as far..

If is a perfect group, then the group center of the quotient group , where is the group center of , is the trivial group.

A group that coincides with its commutator subgroup.If is a non-Abelian group, its commutator subgroup is a normal subgroup other than the trivial group. It follows that if is simple, it must be perfect. The converse, however, is not necessarily true. For example, the special linear group is always perfect if (Rose 1994, p. 61), but if is not a power of 2 (i.e., the field characteristic of the finite field is not 2), it is not simple, since its group center contains two elements: the identity matrix and its additive inverse , which are different because .

An additive group is a group where the operation is called addition and is denoted . In an additive group, the identity element is called zero, and the inverse of the element is denoted (minus ). The symbols and terminology are borrowed from the additive groups of numbers: the ring of integers , the field of rational numbers , the field of real numbers , and the field of complex numbers are all additive groups.In general, every ring and every field is an additive group. An important class of examples is given by the polynomial rings with coefficients in a ring . In the additive group of the sum is performed by adding the coefficients of equal terms,(1)Modules, abstractvector spaces, and algebras are all additive groups.The sum of vectors of the vector space is defined componentwise,(2)and so is the sum of matrices with entries in a ring ,(3)which is part of the -module structure of the set of matrices .Any quotient group of an Abelian additive group is again..

For any prime number and any positive integer , the -rank of a finitely generated Abelian group is the number of copies of the cyclic group appearing in the Kronecker decomposition of (Schenkman 1965). The free (or torsion-free) rank of is the number of copies of appearing in the same decomposition. It can be characterized as the maximal number of elements of which are linearly independent over . Since it is also equal to the dimension of as a vector space over , it is often called the rational rank of . Munkres (1984) calls it the Betti number of .Most authors refer to simply as the "rank" of (Kargapolov and Merzljakov 1979), whereas others (Griffith 1970) use the word "rank" to denote the sum . In this latter meaning, the rank of is the number of direct summands appearing in the Kronecker decomposition of ...

In a monoid or multiplicative group where the operation is a product , the multiplicative inverse of any element is the element such that , with 1 the identity element.The multiplicative inverse of a nonzero number is its reciprocal (zero is not invertible). For complex ,The inverse of a nonzero real quaternion (where are real numbers, and not all of them are zero) is its reciprocalwhere .The multiplicative inverse of a nonsingular matrixis its matrix inverse.To detect the multiplicative inverse of a given element in the multiplication table of finite multiplicative group, traverse the element's row until the identity element 1 is encountered, and then go up to the top row. In this way, it can be immediately determined that is the multiplicative inverse of in the multiplicative group formed by all complex fourth roots of unity...

In an additive group , the additive inverse of an element is the element such that , where 0 is the additive identity of . Usually, the additive inverse of is denoted , as in the additive group of integers , of rationals , of real numbers , and of complex numbers , where The same notation with the minus sign is used to denote the additive inverse of a vector,(1)of a polynomial,(2)of a matrix(3)and, in general, of any element in an abstractvector space or a module.

The identity element of an additive group , usually denoted 0. In the additive group of vectors, the additive identity is the zero vector , in the additive group of polynomials it is the zero polynomial , in the additive group of matrices it is the zero matrix.

The Curry triangle, also sometimes called the missing square puzzle, is a dissection fallacy created by American neuropsychiatrist L. Vosburgh Lions as an example of a phenomenon discovered by Paul Curry. The figure apparently shows that a triangle of area 60, a triangle of area 58 containing a rectangular hole, and a broken rectangle of area 59 can all be formed out of the same set of 6 polygonal pieces. The explanation for this lies in the inaccuracy of the initial subdivision. In the diagrams, the small and large right triangles are similar, hence they cannot have perpendicular sides of lengths and , respectively, as apparently shown in the drawing.

The topology induced by a topological space on a subset . The open sets of are the intersections , where is an open set of .For example, in the relative topology of the interval induced by the Euclidean topology of the real line, the half-open interval is open since it coincides with . This example shows that an open set of the relative topology of need not be open in the topology of .

A Cartesian product of any finite or infinite set of copies of , equipped with the product topology derived from the discrete topology of . It is denoted . The name is due to the fact that for , this set is closely related to the Cantor set (which is formed by all numbers of the interval which admit an expansion in base 3 formed by 0s and 2s only), and this gives rise to a one-to-one correspondence between and the Cantor set, which is actually a homeomorphism. In the symbol denoting the Cantor discontinuum, can be replaced by 2 and by .

A topological space that is not connected, i.e., which can be decomposed as the disjoint union of two nonempty open subsets. Equivalently, it can be characterized as a space with more than one connected component.A subset of the Euclidean plane with more than one element can always be disconnected by cutting it through with a line (i.e., by taking out its intersection with a suitable straight line). In fact, it is certainly possible to find a line such that two points of lie on different sides of . If the Cartesian equation of is(1)for fixed real numbers , then the set is disconnected, since it is the union of the two nonempty open subsets(2)and(3)which are the sets of elements of lying on the two sides of .

A topology defined on a totally ordered set whose open sets are all the finite intersections of subsets of the form or , where .The order topology of the real line is the Euclidean topology. The order topology of is the discrete topology, since for all ,is an open set.

The perspective image of an infinite checkerboard. It can be constructed starting from any triangle , where and form the near corner of the floor, and is the horizon (left figure). If is the corner tile, the lines and must be parallel to and respectively. This means that in the drawing they will meet and at the horizon, i.e., at point and point respectively (right figure). This property, of course, extends to the two bunches of perpendicular lines forming the grid.The adjacent tile (left figure) can then be determined by the following conditions: 1. The new vertices and lie on lines and respectively. 2. The diagonal meets the parallel line at the horizon . 3. The line passes through . Similarly, the corner-neighbor of (right figure) can be easily constructed requiring that: 1. Point lie on . 2. Point lie on the common diagonal of the two tiles. 3. Line pass through . Iterating the above procedures will yield the complete picture. This construction shows..

A path is a continuous mapping , where is the initial point, is the final point, and denotes the space of continuous functions. The notation for a path parametrized by is commonly denoted .A graph path is a sequence such that , , ..., are graph edges of the graph and the are distinct.A path in a topological space is a continuous map . Often the name path is given to the image of .

A functor is said to be faithful if it is injective on maps. This does not necessarily imply injectivity on objects. For example, the forgetful functor from the category of groups to the category of sets is faithful, but it identifies non-isomorphic groups having the same underlying set. Conversely, a functor injective on objects need not be injective on maps. For example, a counterexample is the functor on the category of vector spaces which leaves every vector space unchanged and sends every map to the zero map.A functor which is injective both on objects and maps is sometimes called an embedding.

The nullity of a linear transformation of vector spaces is the dimension of its null space. The nullity and the map rank add up to the dimension of , a result sometimes known as the rank-nullity theorem.

An abstract vector space of dimension over a field is the set of all formal expressions(1)where is a given set of objects (called a basis) and is any -tuple of elements of . Two such expressions can be added together by summing their coefficients,(2)This addition is a commutative group operation, since the zero element is and the inverse of is . Moreover, there is a natural way to define the product of any element by an arbitrary element (a so-called scalar) of ,(3)Note that multiplication by 1 leaves the element unchanged.This structure is a formal generalization of the usual vector space over , for which the field of scalars is the real field and a basis is given by . As in this special case, in any abstract vector space , the multiplication by scalars fulfils the following two distributive laws: 1. For all and all , . 2. For all and all , . These are the basic properties of the integer multiples in any commutative additive group. This special behavior of a product..

A tetramagic cube is a magic cube that remains magicwhen all its numbers are squared, cubed, and taken to the fourth power.Only two tetramagic cubes are known, and both were found by C. Boyer in 2003. The smaller of these is a tetramagic cube of order 1024; this cube, its square, and its cube are perfect, while its fourth power is only semiperfect. The larger is a perfect tetramagic cube of order 8192; this cube, its square, its cube, and its fourth power are perfect.

A bimagic cube is a (normal) magic cube that remains magic when all its elements are squared. Of course, even a normal magic cubic becomes nonnormal (i.e., contains nonconsecutive elements) upon squaring.Cazalas (1934) attempted but failed to construct a bimagic cube (Boyer). David M. Collison apparently constructed a bimagic cube of order 25 in an unpublished paper (Hendricks 1992), but it was not until the year 2000 that John Hendricks published an order 25 perfect magic cube whose square is a semiperfect magic cube.On January 20, 2003, Christian Boyer discovered an order 16 bimagic cube (where the cube itself is perfect magic, but its square is only semiperfect magic). This was rapidly followed by another order 16 bimagic cube (where the base cube is perfect and its square semiperfect) on January 23, an order 32 bimagic cube (where both the base cube and its square are perfect) on January 27, and an order 27 bimagic cube (where the base..

A string figure is any pattern produced when a looped string is spanned between two hands and is twisted and woven in various manners around the fingers and the wrists. The combinations of crossings which can be realized in this way can be studied using knot theory.The string figure above is known as the Apache door (Jayne 1975, pp. 12-15, Fig. 21) or tent flap (Ball 1971, p. 5, Fig. 2).The string figure illustrated above is known as "Jacob's ladder," Osage diamonds (Jayne 1975, pp. 24-27, Fig. 50), the fishing net, or quadruple diamonds (Ball 1971, p. 19, Fig. 7).String figures, which belong to the ancient traditions of many peoples around the world, and are even present in primitive cultures, are nowadays considered as a recreational activity in mathematics education. In English-speaking countries they are also known as the children's game called "cat's cradle."..

A theorem giving a criterion for an origami construction to be flat. Kawasaki's theorem states that a given crease pattern can be folded to a flat origami iff all the sequences of angles , ..., surrounding each (interior) vertex fulfil the following conditionNote that the number of angles is always even; each of them corresponds to a layer of the folded sheet.The rule evidently applies to the case of a rectangular sheet of paper folded twice, where the crease pattern is formed by the bisectors. But there are many more interesting examples where the above property can be checked (see, for example, the crane origami in the above figure).

A curve whose name means skull-like. It is given by the polar equationwhere , , , , and . The top of the curve corresponds to , while the bottom corresponds to .It has area given bywhere is an Appell hypergeometric function.

A surface which can be interpreted as a self-intersecting rectangle in three dimensions. The Whitney umbrella is the only stable singularity of mappings from to . It is given by the parametric equations(1)(2)(3)for . The center of the "plus" shape which is the end of the line of self-intersection is a pinch point. The coefficients of the first fundamental form are(4)(5)(6)and the second fundamental form are(7)(8)(9)giving area element(10)and Gaussian curvature and meancurvature(11)(12)Note that the ruled cubicsurface given by the equation:(13)is the union of Whitney umbrella and the ray , , called the handle of the Whitney umbrella.

The plane figure formed by a sequence of circles , , , ... that are all tangent to each other at the same point and such that the sequence of radii converges to zero. In the figure above, is chosen to be the circle with center and radius .The topology of this set and of its generalizations to higher dimensions has been intensively studied in recent years (Eda 2000, Eda and Kawamura 2002ab). This research was motivated by the following striking discovery: although the fundamental group of the circle is , the fundamental group of the figure eight is , where denotes the free product, and in general the fundamental group of the -petalled rose is (Massey 1989, pp. 123-125), the fundamental group of the Hawaiian ring is not a free group (Higman 1952, de Smit 1992, Black 1996).

A quadrilateral whose consecutive sides have the lengths , , , , where(1)and(2)Brahmagupta's trapezium is a cyclic quadrilateralwith perpendicular diagonals.It has area(3)circumradius,(4)and the diagonal lengths(5)(6)All these values are rational if and are. In particular, if and are Pythagorean triples, the area, circumdiameter, the lengths of the diagonals are all integers.

A combinatorial conjecture formulated by Kneser (1955). It states that whenever the -subsets of a -set are divided into classes, then two disjoint subsets end up in the same class.Lovász (1978) gave a proof based on graph theory. In particular, he showed that the Kneser graph, whose vertices represent the -subsets, and where each edge connects two disjoint subsets, is not -colorable. More precisely, his results says that the chromatic number is equal to , and this implies that Kneser's conjecture is always false if the number of classes is increased to .An alternate proof was given by Bárány (1978).

The singleton set , with respect to the trivial group structure defined by the addition . The element 0 is the additive identity element of the group, and also the additive inverse of itself.The zero group is a minimal example of group, hence it is called a trivial group. Another example of trivial group is the multiplicative group , where .

A group whose group operation is identified with multiplication. As with normal multiplication, the multiplication operation on group elements is either denoted by a raised dot or omitted entirely, giving the notation or . In a multiplicative group, the identity element is denoted 1, and the inverse of the element is written as , voiced " inverse." This notation and terminology is borrowed from the multiplicative groups formed by numbers, where the operation is the usual arithmetical product, the identity element is the number 1, and the inverse coincides with the multiplicative reciprocal.The simplest examples are the trivial group and , the latter of which is isomorphic to the cyclic additive group . The elements of are the square roots of unity, and in general, the set of all complex th roots of unity is a cyclic multiplicative group of order ,(1)where the generator is any primitive th root of unity. These groups are all subgroups..

There are two definitions of a metacyclic group. 1. A metacyclic group is a group such that both its commutator subgroup and the quotient group are cyclic (Rose 1994, p. 247). 2. A group is metacyclic if it has a cyclic normal subgroup such that the quotient group is also cyclic (Rose 1994, p. 56). In general, a group may be metacyclic according to the second definition and fail the first one. For example, the quaternion group has a normal cyclic subgroup of order 4, thus it satisfies definition (2). On the other hand, the commutant consists of two elements , the quotient is isomorphic to the finite group C2×C2, and thus the group is not cyclic.The first definition is more classical, but nowadays essentially all algebraists use the second definition, which is the one used in the remainder of this article.Metacyclic groups are solvable and have a compositionseries of length two.A complete classification of finite metacyclic groups..

A group such that the quotient group , where is the group center of , is Abelian. An equivalent condition is that the commutator subgroup is contained in .

The commutator subgroup (also called a derived group) of a group is the subgroup generated by the commutators of its elements, and is commonly denoted or . It is the unique smallest normal subgroup of such that is Abelian (Rose 1994, p. 59). It can range from the identity subgroup (in the case of an Abelian group) to the whole group. Note that not every element of the commutator subgroup is necessarily a commutator.For instance, in the quaternion group (, , , ) with eight elements, the commutators form the subgroup . The commutator subgroup of the symmetric group is the alternating group. The commutator subgroup of the alternating group is the whole group . When , is a simple group and its only nontrivial normal subgroup is itself. Since is a nontrivial normal subgroup, it must be .The first homology of a group is the Abelianization..

The Grassmannian is the set of -dimensional subspaces in an -dimensional vector space. For example, the set of lines is projective space. The real Grassmannian (as well as the complex Grassmannian) are examples of manifolds. For example, the subspace has a neighborhood . A subspace is in if and and . Then for any , the vectors and are uniquely determined by requiring and . The other six entries provide coordinates for .In general, the Grassmannian can be given coordinates in a similar way at a point . Let be the open set of -dimensional subspaces which project onto . First one picks an orthonormal basis for such that span . Using this basis, it is possible to take any vectors and make a matrix. Doing this for the basis of , another -dimensional subspace in , gives a -matrix, which is well-defined up to linear combinations of the rows. The final step is to row-reduce so that the first block is the identity matrix. Then the last block is uniquely determined by ...

The kernel of a module homomorphism is the set of all elements of which are mapped to zero. It is the kernel of as a homomorphism of additive groups, and is a submodule of .

The identity element of an additive monoid or group or of any other algebraic structure (e.g., ring, module, abstract vector space, algebra) equipped with an addition. It is also called the additive identity and is denoted 0. The name and the symbol are borrowed from the ring of integers whose additive identity is, of course, number 0.The zero element of a ring has the property that for all and, moreover, for every element of an -module it holds that . Here, the indices distinguish the zero element of the ring from the zero element of the module. The latter also fulfils the rule for all . The notation 0 is sometimes also used for the universal bound of a Boolean algebra . In fact it behaves with respect to the operation like a zero element with respect to multiplication, since for all ...

A module over a unit ring is called divisible if, for all which are not zero divisors, every element of can be "divided" by , in the sense that there is an element in such that . This condition can be reformulated by saying that the multiplication by defines a surjective map from to .It can be shown that every injective -module is divisible, but the converse only holds for particular classes of rings, e.g., for principal ideal domains. Since and are evidently divisible -modules, this allows us to conclude that they are also injective.An additive Abelian group is called divisible if it is so as a -module.

A module having only one element: the singleton set . It is a module over any ring with respect to the multiplication defined by(1)for every , and the addition(2)which makes it a trivial additive group. The only element is, in particular, its zero element. Therefore, a trivial module is often called the zero module, and written as .The notion of trivial module is a special case of the more general notion of trivial module structure, which can be defined on every additive Abelian group with respect to every ring by setting(3)for all and all .

An operation on rings and modules. Given a commutative unit ring , and a subset of , closed under multiplication, such that , and , the localization of at is the ring(1)where the addition and the multiplication of the formal fractions are defined according to the natural rules,(2)and(3)The ring is a subring of via the identification .For an -module , the localization of at is defined as the tensor product , i.e., as the set of linear combinations of the elementary tensors(4)which are also denoted for short.The properties required for the subset are fulfilled by 1. The set of non zero-divisors of ; in this case is the ring of fractions of . 2. The complement of any prime ideal of : in this case the clumsy notation is replaced by . This ring is called the localization of at , and it is a local ring, with maximal ideal . The name given to this operation derives from the geometric meaning it takes when applied to the rings associated with algebraic varieties.The union..

A proving technique in homological algebra which consists in looking for equivalent map compositions in commutative diagrams, and in exploiting the properties of injective, surjective and bijective homomorphisms and of exact sequences.The construction of the connecting homomorphism in the proof of the snake lemma is an example of diagram chasing.

A submodule of a module such that for any other nonzero submodule of , the intersection is not the zero module. is also called an essential submodule of , whereas is called an essential extension of .

The descending chain condition, commonly abbreviated "D.C.C.," is the dual notion of the ascending chain condition. The descending chain condition for a partially ordered set requires that all decreasing sequences in become eventually constant.A module fulfilling the descending chain condition iscalled Artinian.

A submodule of a module that is not the intersection of two submodules of in which it is properly contained. In other words, for all submodules and of ,Using a less common terminology, this is equivalent to requiring that the quotient module be meet-irreducible.

A nonzero module over a ring whose only submodules are the module itself and the zero module. It is also called a simple module, and in fact this is the name more frequently used nowadays (Rowen, 1988). Behrens' (1972, p. 23) definition includes the additional condition that be not the zero module.Sometimes, the term irreducible is used as an abbreviation for meet-irreducible (Kasch 1982), which means that the intersection of two nonzero submodules is always nonzero.These two irreducibility notions are different: every irreducible module is meet-irreducible, but the converse does not hold. For example, the submodules of are , and , so is not irreducible, whereas it is certainly meet-irreducible.This ambiguity in terminology is solved in the context of rings, since a simple ring is a ring that is irreducible as a module over itself, whereas an irreducible ring is a ring which is meet-irreducible as a module over itself.Irreducible modules..

For every module over a unit ring , the tensor product functor is a covariant functor from the category of -modules to itself. It maps every -module to and every module homomorphism to the module homomorphismdefined byThe tensor product functor is defined similarly.

An injective module is the dual notion to the projective module. A module over a unit ring is called injective iff whenever is contained as a submodule in a module , there exists a submodule of such that the direct sum is isomorphic to (in other words, is a direct summand of ). The subset of is an example of a noninjective -module; it is a -submodule of , and it is isomorphic to ; , however, is not isomorphic to the direct sum . The field of rationals and its quotient module are examples of injective -modules.A direct product of injective modules is always injective. The corresponding property for direct sums does not hold in general, but it is true for modules over Noetherian rings.The notion of injective module can also be characterized by means of commutative diagrams, split exact sequences, or exact functors...

A non-zero module which is not the direct sum of two of its proper submodules. The negation of indecomposable is, of course, decomposable. An abstract vector space is indecomposable iff it has dimension 1.As a consequence of Kronecker basis theorem, an Abelian group is indecomposable iff it is either isomorphic to or to , where is a prime power. This is not the case for , and in fact we have

The cokernel of a group homomorphism of Abelian groups (modules, or abstract vector spaces) is the quotient group (quotient module or quotient space, respectively) .

A module having dual properties with respect to a freemodule, as enumerated below. 1. Every free module is projective; every cofree module is injective. 2. For every module , there is a surjective homomorphism from a free module to ; for every module , there is an injective homomorphism from to a cofree module. 3. A module is projective iff it can be completed by a direct sum to a free module; a module is injective iff it can be completed by a direct product to a cofree module. Every cofree module over a unit ring is isomorphic to a direct productindexed on some set .

The notion of height is defined for proper ideals in a commutative Noetherian unit ring . The height of a proper prime ideal of is the maximum of the lengths of the chains of prime ideals contained in ,The height of any proper ideal is the minimum of the heights of the prime ideals containing .

A coequalizer of a pair of maps in a category is a map such that 1. , where denotes composition. 2. For any other map with the same property, there is exactly one map such that i.e., one has the above commutative diagram. It can be shown that the coequalizer is an epimorphismand that, moreover, it is unique up to isomorphism.In the category of sets, the coequalizer is given bythe quotient setand by the canonical map , where is the minimal equivalence relation on that identifies and for all .The same construction is valid in the categories of additive groups, modules, and vector spaces. In these cases, the cokernel of a morphism can be viewed, in a more abstract categorical setting, as the coequalizer of and the zero map.The dual notion of the coequalizer is the equalizer.

Given a short exact sequence of modules(1)let(2)(3)be projective resolutions of and , respectively. Then there is a projective resolution of (4)such that the above diagrams are commutative. Here, is the injection of the first summand, whereas is the projection onto the second factor for .The name of this lemma derives from the shape of the diagram formed by the shortexact sequence and the given projective resolutions.

In a cochain complex of modulesthe module of -cocycles is the kernel of , which is a submodule of .

A short exact sequence of groups(1)is called split if it essentially presents as the direct sum of the groups and .More precisely, one can construct a commutative diagram as diagrammed above, where is the injection of the first summand and is the projection onto the second summand , and the vertical maps are isomorphisms.Not all short exact sequences of groups are split. For example the short exact sequence diagrammed above cannot be split, since and are non isomorphic finite groups. Note that this is also a short exact sequence of -modules: this shows that being split is a distinguished property of short exact sequences also in the category of modules. In fact, it is related to particular classes of modules.Given a module over a unit ring , all short exact sequences(2)are split iff is projective, and all short exact sequences(3)are split iff is injective.A short exact sequence of vectorspaces is always split...

In a chain complex of modulesthe module of -cycles is the kernel of , which is a submodule of .

In a chain complex of modulesthe module of -boundaries is the image of . It is a submodule of and is contained in the module of -cycles , which is the kernel of .The complex is called exact at if .In the chain complexwhere all boundary operators are the multiplication by 4, for all the module of -boundaries is , whereas the module of -cycles is .

A diagram lemma which states that the above commutative diagram of Abelian groups and group homomorphisms with exact rows gives rise to an exact sequenceThis commutative diagram shows how the first commutative diagram (shown here in blue) can be modified to exhibit the long exact sequence (shown here in red) explicitly. The map is called a connecting homomorphism and describes a curve from the end of the upper row () to the beginning of the lower row (), which suggested the name given to this lemma.The snake lemma is explained in the first scene of Claudia Weill's film Itis My Turn (1980), starring Jill Clayburgh and Michael Douglas.

In a cochain complex of modulesthe module of -coboundaries is the image of . It is a submodule of and is contained in the module of -cocycles .The cochain complex is called exact at if .In the right complex of -modulesfor all , the th module is , and the th coboundary operator maps every element of to the residue class of in . The module of -coboundaries is the set of the residue classes of 0 and in , and the module of -cocycles the set of the residues classes of all even numbers .

A submodule of a module such that for any proper submodule of , the submodule generated by is not the entire module . is also called superfluous submodule.

A short exact sequence of groups , , and is given by two maps and and is written(1)Because it is an exact sequence, is injective, and is surjective. Moreover, the group kernel of is the image of . Hence, the group can be considered as a (normal) subgroup of , and is isomorphic to .A short exact sequence is said to split if there is a map such that is the identity on . This only happens when is the direct product of and .The notion of a short exact sequence also makes sense for modules and sheaves. Given a module over a unit ring , all short exact sequences(2)are split iff is projective, and all short exact sequences(3)are split iff is injective.A short exact sequence of vector spaces is alwayssplit.

Serre's problem, also called Serre's conjecture, asserts that the implication "free module projective module" can be reversed for every module over the polynomial ring , where is a field (Serre 1955).The hard part of the proof, the one concerning finitely generated modules, was given simultaneously, and independently, by D. Quillen in Cambridge, Massachusetts and A. A. Suslin in Leningrad (St. Petersburg) in 1976. As a result, the statement is often referred to as the "Quillen-Suslin theorem."The solution to this difficult problem is part of the work for which Quillen wasawarded the Fields Medal in 1978.Quillen and Suslin received, for other contributions in algebra, the ColePrize in 1975 and 2000 respectively.

Given two modules and over a unit ring , denotes the set of all module homomorphisms from to . It is an -module with respect to the addition of maps,(1)and the product defined by(2)for all . denotes the covariant functor from the category of -modules to itself which maps every module to , and maps every module homomorphism(3)to the module homomorphism(4)such that, for every ,(5)A similar definition is given for the contravariant functor , which maps to and maps to(6)where, for every ,(7)

Given a finitely generated -graded module over a graded ring (finitely generated over , which is an Artinian local ring), define the Hilbert function of as the map such that, for all ,(1)where denotes the length. If is the dimension of , then there exists a polynomial of degree with rational coefficients (called the Hilbert polynomial of ) such that for all sufficiently large .The power series(2)is called the Hilbert series of . It is a rational function that can be written in a unique way in the form(3)where is a finite linear combination with integer coefficients of powers of and . If is positively graded, i.e., for all , then is an ordinary polynomial with integer coefficients in the variable . If moreover , then , i.e., the Hilbert series is a polynomial.

Given a commutative unit ring , and an -module , a sequence of elements of is called a regular sequence for (or an -sequence for short), if, for all , 1. The multiplication by is injective on . 2. . If only condition (1) is fulfilled, the sequence is called weakly regular. An -sequence is usually simply called a regular sequence.

Given a finitely generated -graded module over a graded ring (finitely generated over , which is an Artinian local ring), the Hilbert function of is the map such that, for all ,(1)where denotes the length. If is the dimension of , then there exists a polynomial of degree with rational coefficients (called the Hilbert polynomial of ) such that for all sufficiently large .The power series(2)is called the Hilbert series of . It is a rational function that can be written in a unique way in the form(3)where is a finite linear combination with integer coefficients of powers of and . If is positively graded, i.e., for all , then is an ordinary polynomial with integer coefficients in the variable . If moreover , then , i.e., the Hilbert series is a polynomial.If has a finite graded free resolution(4)then(5)Moreover, if is a regular sequence over of homogeneous elements of degree 1, then the Hilbert function of the -dimensional quotient module is(6)and in particular,(7)These..

The Cartesian product of a countable infinity of copies of the interval . It can be denoted or , where and are the first infinite cardinal and ordinal, respectively. It is homeomorphic to the product space of any countable infinity of closed bounded positive-length intervals.According to another interesting description (Cullen 1968, pp. 164-165), the Hilbert cube can be identified up to homeomorphisms with the metric space formed by all sequences of real numbers such that for all , where the metric is defined asIt is then a subspace of the metric space called a Hilbert space which is formed by all real sequences such that the series converges.The Hilbert cube can be used to characterize classes of topologicalspaces. 1. A topological space that is second countable and T4 is homeomorphic to a subspace of the Hilbert cube. 2. A topological space that is separable andmetrizable is homeomorphic to a subspace of the Hilbert cube. Other statements..

Given a module over a commutative unit ring and a filtration(1)of ideals of , the Rees module of with respect to is(2)which is the set of all formal polynomials in the variable in which the coefficient of is of the form , where and . It is a graded module over the Rees ring .The subscript distinguishes it from the so-called extended Rees module, defined as(3)where for all . This module includes all polynomials containing negative powers of .If is a proper ideal of , the notation (or ) indicates the (extended) Rees module of with respect to the -adic filtration.

An equation representing a locus in the -dimensional Euclidean space. It has the form(1)where the left-hand side is some expression of the Cartesian coordinates , ..., . The -tuples of numbers fulfilling the equation are the coordinates of the points of .For example, the locus of all points in the Euclidean plane lying at distance 1 from the origin is the circle that can be represented using the Cartesian equation(2)Similarly, the locus of all points of the three-dimensional Euclidean space lying at distance 1 from the origin is a sphere of radius 1 centered at the origin can be represented using the Cartesian equation(3)Often the letters , , are used instead of indexed coordinates , , .The intersection of two loci and is the set of points whose coordinates fulfil the system of equations(4)(5)For example, the system(6)(7)represents the intersection of the coordinate plane (the set of points for which ) with the coordinate plane (the set of points..

A minimal free resolution of a finitely generated graded module over a commutative Noetherian -graded ring in which all maps are homogeneous module homomorphisms, i.e., they map every homogeneous element to a homogeneous element of the same degree. It is usually written in the form(1)where indicates the ring with the shifted graduation such that, for all ,(2)For all nonnegative integers and all integers , is the number of copies of appearing in the th module of the resolution, and is called graded Betti number. The ordinary th Betti number is .For example, if is the polynomial ring over a field , with the usual graduation, the graded free resolution of is(3)In , the constant polynomials have degree 2. It follows that has degree 5. Similarly, has degree 5 in .The graded free resolution can be used to compute the Hilbertfunction...

If is a submodule of the module over the ring , the quotient group has a natural structure of -module with the product defined byfor all and all .

The free module of rank over a nonzero unit ring , usually denoted , is the set of all sequences that can be formed by picking (not necessarily distinct) elements , , ..., in . The set is a particular example of the algebraic structure called a module since is satisfies the following properties. 1. It is an additive Abelian group with respectto the componentwise sum of sequences,(1)2. One can multiply any sequence with any element of according to the rule(2)and this product fulfils both the associative and the distributive law. The term free module extends to all modules which are isomorphic to , i.e., which have essentially the same structure as . Note that not all modules are free. For example, the quotient ring , where is an integer greater than 1 is not free, since it is a -module having elements, and therefore it cannot be isomorphic to any of the modules , which are all infinite sets. Hence it is not free as a -module, while, of course, it is free as a module..

Baer's criterion, also known as Baer's test, states that a module over a unit ring is injective iff every module homomorphism from an ideal of to can be extended to a homomorphism from to .

A proof that is only based on visual elements, without any comments.An arithmetic identity can be demonstrated by a picture showing a self-evident equality between numerical quantities. The above figure shows that the difference between the th pentagonal number and is equal to three times the th triangular number. Of course, the situation depicted is a particular case of the formula (here it corresponds to ), but it is presented in a way that can be immediately generalized.Another form of proof without words frequently used in elementary geometry is thedissection proof.

A module over a unit ring is called flat iff the tensor product functor (or, equivalently, the tensor product functor ) is an exact functor.For every -module, obeys the implicationwhich, in general, cannot be reversed.A -module is flat iff it is torsion-free: hence and the infinite direct product are flat -modules, but they are not projective. In fact, over a Noetherian ring or a local ring, flatness implies projectivity only for finitely generated modules. This property, together with Serre's problem, allows it to be concluded that the three above implications are equivalences if is a finitely generated module over a polynomial ring , where is a field.

Given a commutative unit ring and a filtration(1)of ideals of , the associated graded ring of with respect to is the graded ring(2)The addition is defined componentwise, and the product is defined as follows. If is the residue class of mod , and is the residue class of mod , then is the residue class of mod . is a quotient ring of the Rees ring of with respect to ,(3)If is a proper ideal of , then the notation indicates the associated graded ring of with respect to the -adic filtration of ,(4)If is Noetherian, then is as well. Moreover is finitely generated over . Finally, if is a local ring with maximal ideal , then(5)

A projective module generalizes the concept of the free module. A module over a nonzero unit ring is projective iff it is a direct summand of a free module, i.e., of some direct sum . This does not imply necessarily that itself is the direct sum of some copies of . A counterexample is provided by , which is a module over the ring with respect to the multiplication defined by . Hence, while a free module is obviously always projective, the converse does not hold in general. It is true, however, for particular classes of rings, e.g., if is a principal ideal domain, or a polynomial ring over a field (Quillen and Suslin 1976). This means that, for instance, is a nonprojective -module, since it is not free.A direct sum of projective modules is always projective, but this property does not apply to direct products. For example, the infinite direct product is not a projective -module.According to its formal definition, a module is projective if, whenever is a quotient..

A filtration of ideals of a commutative unit ring is a sequence of idealssuch that for all indices . An example is the -adic filtration associated with a proper ideal of ,A ring equipped with a filtration is called a filteredring.

Given a module over a commutative unit ring and a filtration(1)of ideals of , the associated graded module of with respect to is(2)which is a graded module over the associated graded ring with respect to the addition and the multiplication by scalars defined componentwise.If is a proper ideal of , then the notation indicates the associated graded module of with respect to the -adic filtration of ,(3)

The ascending chain condition, commonly abbreviated "A.C.C.," for a partially ordered set requires that all increasing sequences in become eventually constant.A module fulfils the ascending chain condition if its set of submodules obeys the condition with respect to inclusion. In this case, is called Noetherian.

A module over a unit ring is called faithfully flat if the tensor product functor is exact and faithful.A faithfully flat module is always flat and faithful, but the converse does not hold in general. For example, is a faithful and flat -module, but it is not faithfully flat: in fact reduces all the quotient modules (and the maps between them) to zero, since for all and all :

A diagram lemma also known as lemma. According to its most general statement, the commutative diagram illustrated above with exact rows and columns can be completed by two morphismswithout losing commutativity.Moreover, the short exact sequenceis exact.The lemma is also true if the roles of the first and the third row are interchanged.

A module over a unit ring is called faithful if for all distinct elements , of , there exists such that . In other words, the multiplications by and by define two different endomorphisms of .This condition is equivalent to requiring that whenever , , one has that for some , i.e., , so that the annihilator of is reduced to . This shows, in particular, that any torsion-free module is faithful. Hence the field of rationals and the polynomial rings are faithful -modules.More generally, any ring containing as a subring is faithful as a module over , since 1 is annihilated only by 0.The -modules are not faithful, since they are annihilated by . In general, a finite module over an infinite ring cannot be faithful, since in this case the infinitely many elements of the ring have to give rise to only a finite number of module endomorphisms...

In a set equipped with a binary operation called a product, the multiplicative identity is an element such thatfor all . It can be, for example, the identity element of a multiplicative group or the unit of a unit ring. In both cases it is usually denoted 1. The number 1 is, in fact, the multiplicative identity of the ring of integers and of its extension rings such as the ring of Gaussian integers , the field of rational numbers , the field of real numbers , and the field of complex numbers . The residue class of number 1 is the multiplicative identity of the quotient ring of for all integers .If is a commutative unit ring, the constant polynomial 1 is the multiplicative identity of every polynomial ring .In a Boolean algebra, if the operation is considered as a product, the multiplicative identity is the universal bound . In the power set of a set , this is the total set .The unique element of a trivial ring is simultaneously the additive identity and multiplicative..

An object is unique if there is no other object satisfying its defining properties. An object is said to be essentially unique if uniqueness is only referred to the underlying structure, whereas the form may vary in ways that do not affect the mathematical content. For the sake of precision, the decomposition of a positive integer into prime factors is not strictly unique, but rather is essentially unique, because it is unique only up to insignificant formal modifications such as permutations of the factors () or changes of sign (). Similarly, the group of order 2 is essentially unique--despite the evidence that the additive group and the multiplicative group are different--because they are isomorphic groups, which differ only in the names given to their elements and their operations.

A one-person game played on a rectangular lattice of lamps which can be turned on and off. A move consists of flipping a "switch" inside one of the squares, thereby toggling the on/off state of this and all four vertically and horizontally adjacent squares. Starting from a randomly chosen light pattern, the aim is to turn all the lamps off. The problem of determining if it is possible to start from set of all lights being on to all lights being off is known as the "all-ones problem." As shown by Sutner (1989), this is always possible for a square lattice (Rangel-Mondragon).This can be translated into the following algebraic problem. 1. Each lamp configuration can be viewed as a matrix with entries in (i.e., a (0,1)-matrix, where each 1 represents a burning light and 0 represents a light turned off. For example, for the case,(1)2. The action of the switch placed at can be interpreted as the matrix addition , where is the matrix in which..

An ascending chain of subspaces of a vector space. If is an -dimensional vector space, a flag of is a filtration(1)where all inclusions are strict. Hence(2)so that . If equality holds, then for all , and the flag is called complete or full. In this case it is a composition series of .A full flag can be constructed by fixing a basis of , and then taking for all .A flag of any length can be obtained from a full flag by taking out some of the subspaces. Conversely, every flag can be completed to a full flag by inserting suitable subspaces. In general, this can be done in different ways. The following flag of (3)can be completed by switching in any line of the -plane passing through the origin. Two different full flags are, for example,(4)and(5)Schubert varieties are projective varieties definedfrom flags...

The orthogonal complement of a subspace of the vector space is the set of vectors which are orthogonal to all elements of . For example, the orthogonal complement of the space generated by two non proportional vectors , of the real space is the subspace formed by all normal vectors to the plane spanned by and .In general, any subspace of an inner product space has an orthogonal complement andThis property extends to any subspace of a space equipped with a symmetric or differential -form or a Hermitian form which is nonsingular on .

The semigroup algebra , where is a field and a semigroup, is formally defined in the same way as the group algebra . Similarly, a semigroup ring is a variation of the group ring , where the group is replaced by a semigroup . Usually, it is required that have an identity element so that is a unit ring and is a subring of .The group algebra is the set of all formal expressions(1)where for all and for all but finitely many indices so that for sufficiently large (say, ). Hence, we can write the general element as(2)Assigning(3)defines an isomorphism of -algebras between and the polynomial ring .More generally, if is the subsemigroup of generated by the elements , for , the semigroup algebra is isomorphic to the subalgebra of the polynomial ring generated by the monomials

A topology on a set whose open sets are the unions of open ballswhere is a pseudometric on , is any point of , and .There is a remarkable difference between a metric and a pseudometric topology. The former is always , whereas the latter is, in general, not even . In fact, a pseudometric allows for some distinct points and , and then every open ball containing contains and conversely, so that no open set can separate the two points.

The symbol , used for the intersection of sets, and sometimes also for the logical connective AND instead of the symbol (wedge). In fact, for any two sets and and this equivalence demonstrates the connection between the set-theoretical and the logical meaning.The term "cap" is also used to refer to the topological object produced by puncturing a surface a single time, attaching two zips around the puncture in opposite directions, distorting the hole so that the zips line up, and then zipping up. The cap is topologically trivial in the sense that a surface with a cap is topologically equivalent to a surface without one.

The topology on the Cartesian product of two topological spaces whose open sets are the unions of subsets , where and are open subsets of and , respectively.This definition extends in a natural way to the Cartesian product of any finite number of topological spaces. The product topology ofwhere is the real line with the Euclidean topology, coincides with the Euclidean topology of the Euclidean space .In the definition of product topology of , where is any set, the open sets are the unions of subsets , where is an open subset of with the additional condition that for all but finitely many indices (this is automatically fulfilled if is a finite set). The reason for this choice of open sets is that these are the least needed to make the projection onto the th factor continuous for all indices . Admitting all products of open sets would give rise to a larger topology (strictly larger if is infinite), called the box topology.The product topology is also called..

The term cube is used in topology to denote the Cartesian product of any (finite or infinite) number of copies of the closed interval equipped with the product topology derived from the relative topology induced on each interval by the Euclidean topology of the real line. A particular type is the Hilbert cube.

In a space equipped with a symmetric, differential -form, or Hermitian form, the orthogonal sum is the direct sum of two subspaces and , which are mutually orthogonal. It is denoted .More generally,denotes a direct sum of subspaces of which are pairwise orthogonal.

In elementary geometry, orthogonal is the same as perpendicular. Two lines or curves are orthogonal if they are perpendicular at their point of intersection. Two vectors and of the real plane or the real space are orthogonal iff their dot product . This condition has been exploited to define orthogonality in the more abstract context of the -dimensional real space .More generally, two elements and of an inner product space are called orthogonal if the inner product of and is 0. Two subspaces and of are called orthogonal if every element of is orthogonal to every element of . The same definitions can be applied to any symmetric or differential k-form and to any Hermitian form.

A property that is always fulfilled by the product of topological spaces, if it is fulfilled by each single factor. Examples of productive properties are connectedness, and path-connectedness, axioms , , and , regularity and complete regularity, the property of being a Tychonoff space, but not axiom and normality, which does not even pass, in general, from a space to . Metrizability is not productive, but is preserved by products of at most spaces. Separability is not productive, but is preserved by products of at most spaces.Compactness is productive by the Tychonoff theorem.

According to most authors (e.g., Kelley 1955, p. 113; McCarty 1967, p. 144; Willard 1970, p. 92) a regular space is a topological space in which every neighborhood of a point contains a closed neighborhood of the same point.Another equivalent condition is the following: for every closed set and every point there are two disjoint open sets and such that and .In other sources (e.g., Bourbaki 1989, p. 80; Cullen 1968, p. 113) regularity is defined differently, using separation axioms.

Also called a -space, a Tychonoff space is a completely regular space (with the additional condition that it be for those authors who do not assume this in the definition of completely regular.) In any case, Tychonoff spaces can be characterized as the topological spaces which are homeomorphic to a subspace of some cube (equivalently, which are subspaces of some compact T2-space, or some T4-space.)

A Tychonoff plank is a topological space that is an example of a normal space which has a non-normal subset, thus showing that normality is not a hereditary property. Let be the set of all ordinals which are less than or equal to , and the set of all ordinals which are less than or equal to . Consider the set with the product topology induced by the order topologies of and . Then is normal, but the subset is not. It can be shown that the set of all elements of whose first coordinate is equal to and the set of all elements of whose second coordinate is equal to are disjoint closed subsets , but there are no disjoint open subsets and of such that and .

A topological space having a countable dense subset. An example is the Euclidean space with the Euclidean topology, since it has the rational lattice as a countable dense subset and it is easy to show that every open -ball contains a point whose coordinates are all rational.

A class formed by sets in which have essentially the same structure, regardless of size, shape and dimension. The "essential structure" is what a set keeps when it is transformed by compressing or dilating its parts, but without cutting or gluing. The most important feature that is preserved is the system of internal closed paths. In particular, the fundamental group remains unchanged. This object, however, only characterizes the loops, i.e., the paths which are essentially circular lines, whereas the homotopy type also refers to higher dimensional closed paths, which correspond to the boundaries of -spheres. Hence the homotopy type yields a more precise classification of geometric objects. As for the circular paths, it makes no difference whether the object is located in the plane or on the surface of a sphere, so the fundamental group is the same in both cases.The homotopy type, however, is different, since the plane does not..

A method for verifying the correctness of an arithmetical operation on natural numbers, based on the same principle as casting out nines. The methods of sevens takes advantage of the fact that the residue (mod 7) of a sum (or product) must be equal to the sum (or product) of the residues of the summands (or factors).For example, the correct sum(1)corresponds to a correct sum of residues mod 7(2)where, on the right-hand side, 9 has been replaced by its residue 2 (mod 7).On the other hand, the incorrect sum(3)gives rise to an incorrect sum of residues(4)since the right-hand side should be 0.Tests based on the comparison of residues are not completely reliable since they leave some errors undetected (namely, an incorrect sum can produce a correct sum of residues). Hence it can be helpful to double-check with respect to 7 and 9...

If is a sentential formula depending on a variable ranging in a set of real numbers, the sentence(1)means(2)An example is the proposition(3)which is true, since the inequality is fulfilled for .The statement can also be rephrased as follows: the terms of the sequence become eventually smaller than 0.0001.There are various mathematical jokes involving "sufficiently large." For example, " for sufficiently large values of 1" and "this feature will ship in version 1.0 for sufficiently large values of 1."

The Euclidean plane parametrized by coordinates, so that each point is located based on its position with respect to two perpendicular lines, called coordinate axes. They are two copies of the real line, and the zero point lies at their intersection, called the origin. The coordinate axes are usually called the x-axis and y-axis, depicted above. Point is associated with the coordinates corresponding to its orthogonal projections onto the -axis and the -axis respectively.

Cartesian coordinates are rectilinear two- or three-dimensional coordinates (and therefore a special case of curvilinear coordinates) which are also called rectangular coordinates. The two axes of two-dimensional Cartesian coordinates, conventionally denoted the x- and y-axes (a notation due to Descartes), are chosen to be linear and mutually perpendicular. Typically, the -axis is thought of as the "left and right" or horizontal axis while the -axis is thought of as the "up and down" or vertical axis. In two dimensions, the coordinates and may lie anywhere in the interval , and an ordered pair in two-dimensional Cartesian coordinates is often called a point or a 2-vector.The three-dimensional Cartesian coordinate system is a natural extension of the two-dimensional version formed by the addition of a third "in and out" axis mutually perpendicular to the - and -axes defined above. This new axis is conventionally..

A plane coordinate system whose axes are not perpendicular. The -coordinate of a point is the abscissa of its projection onto the -axis in the direction of the -axis, and the -coordinate is similarly determined.Unlike in the more general case of affine coordinates, in oblique coordinates, the unit length is the same on both axes. If the Cartesian equation of a curve in a Cartesian coordinate system is applied to an oblique coordinate system, the result will be a distorted curve. For example, a circle will be transformed into an ellipse.

The coordinates representing any point of an -dimensional affine space by an -tuple of real numbers, thus establishing a one-to-one correspondence between and .If is the underlying vector space, and is the origin, every point of is identified with the -tuple of the components of vector with respect to a given basis of .If is a three-dimensional space, each basis can be depicted by choosing its elements as the unit vectors of the -axis, the -axis, and the -axis, respectively. In general, this will produce three axes which are not necessarily perpendicular, and where the units are set differently. Hence, Cartesian coordinates are a very special kind of affine coordinates that correspond to the case where , , .

The adjective "affine" indicates everything that is related to the geometry of affine spaces. A coordinate system for the -dimensional affine space is determined by any basis of vectors, which are not necessarily orthonormal. Therefore, the resulting axes are not necessarily mutually perpendicular nor have the same unit measure. In this sense, affine is a generalization of Cartesian or Euclidean.An example of an affine property is the average area of a random triangle chosen inside a given triangle (i.e., triangle triangle picking). Because this problem is affine, the ratio of the average area to the original triangle is a constant independent of the actual triangle chosen. Another example of an affine property is the areas (relative to the original triangle) of the regions created by connecting the side -multisectors of a triangle with lines drawn to the opposite vertices (i.e., Marion's theorem).An example of a property that..

The dual of Pascal's theorem (Casey 1888, p. 146). It states that, given a hexagon circumscribed on a conic section, the lines joining opposite polygon vertices (polygon diagonals) meet in a single point.In 1847, Möbius (1885) gave a statement which generalizes Brianchon's theorem: if all lines (except possibly one) connecting two opposite vertices of a ()-gon circumscribed on a conic section meet in one point, then the same is true for the remaining line.

The perpendicular bisector of a linesegment is the locus of all points that are equidistant from its endpoints.This theorem can be applied to determine the center of a given circle with straightedge and compass. Pick three points , and on the circle. Since the center is equidistant from all of them, it lies on the bisector of segment and also on the bisector of segment , i.e., it is the intersection point of the two bisectors. This construction is shown on a window pane by tutor Justin McLeod (Mel Gibson) to his pupil Chuck Norstadt (Nick Stahl) in the 1993 film The Man Without a Face.

A geometric implement discovered in a 19th century book, and whose inventor is unknown. It essentially consists of a semicircle, a segment which prolongs its diameter and is equal to the radius, and a segment perpendicular to it.It can be used to trisect an angle, an operation impossible with straightedge and compass. If it is adjusted to an the angle so that 1. The line passes through . 2. lies on line . 3. The line is tangent to the semicircle. Then angle is equal to one third of .The interesting fact about the tomahawk is that it can be easily constructed with straightedge and compass. Hence these tools are, from a merely practical point of view, sufficient to trisect an angle. This does not contradict what is known from mathematical theory, since the procedure of shifting a figure on the paper until its parts fall in given positions is not an Euclidean construction...

The trivial group, denoted or , sometimes also called the identity group, is the unique (up to isomorphism) group containing exactly one element , the identity element. Examples include the zero group (which is the singleton set with respect to the trivial group structure defined by the addition ), the multiplicative group (where ), the point group , and the integers modulo 1 under addition. When viewed as a permutation group on letters, the trivial group consists of the single element which fixes each letter.The trivial group is (trivially) Abelian and cyclic.The multiplication table for is given below. 111The trivial group has the single conjugacy class and the single subgroup .

Every finite Abelian group can be written as a group direct product of cyclic groups of prime power group orders. In fact, the number of nonisomorphic Abelian finite groups of any given group order is given by writing aswhere the are distinct prime factors, thenwhere is the partition function. This gives 1, 1, 1, 2, 1, 1, 1, 3, 2, ... (OEIS A000688).More generally, every finitely generated Abelian group is isomorphic to the group direct sum of a finite number of groups, each of which is either cyclic of prime power order or isomorphic to . This extension of Kronecker decomposition theorem is often referred to as the Kronecker basis theorem.

A generalization of the Kronecker decomposition theorem which states that every finitely generated Abelian group is isomorphic to the group direct sum of a finite number of groups, each of which is either cyclic of prime power order or isomorphic to . This decomposition is unique, and the number of direct summands is equal to the group rank of the Abelian group.

Given a ring with identity, the special linear group is the group of matrices with elements in and determinant 1.The special linear group , where is a prime power, the set of matrices with determinant and entries in the finite field . is the corresponding set of complex matrices having determinant . is a subgroup of the general linear group and is a Lie-type group. Both and are genuine Lie groups.

The function from a given nonempty set to the power set that maps every element of to the set .

A map between topological spaces that maps closed sets to closed sets. If is bijective, thenwhere denotes the inverse map. In particular, a homeomorphism can be characterized as a continuous bijection which is open (or, equivalently, closed).

A polynomial admitting a multiplicative inverse. In the polynomial ring , where is an integral domain, the invertible polynomials are precisely the constant polynomials such that is an invertible element of . In particular, if is a field, the invertible polynomials are all constant polynomials except the zero polynomial.If is not an integral domain, there may be in invertible polynomials that are not constant. In , for instance, we have:which shows that the polynomial is invertible, and inverse to itself.

Given a commutative unit ring and a filtration(1)of ideals of , the Rees ring of with respect to is(2)which is the set of all formal polynomials in the variable in which the coefficient of lies in . It is a graded ring with respect to the usual addition and multiplication of polynomials, which makes it a subring of the polynomial ring . It is also a subring of the extended Rees ring(3)which is a subring of , the ring of all finite linear combinations of integer (possibly negative) powers of .If is a proper ideal of , the notation (or ) indicates the (extended) Rees ring of with respect to the -adic filtration of . If is the polynomial ring over a field , then is the coordinate ring of the blow-up of the affine space along the affine variety .

Any ideal of a ring which is strictly smaller than the whole ring. For example, is a proper ideal of the ring of integers , since .The ideal of the polynomial ring is also proper, since it consists of all multiples of , and the constant polynomial 1 is certainly not among them.In general, an ideal of a unit ring is proper iff . The latter condition is obviously sufficient, but it is also necessary, because would imply that for all ,so that , a contradiction.Note that the above condition follows by definition: an ideal is always closed under multiplication by any element of the ring. The same property implies that an ideal containing an invertible element cannot be proper, because , where denotes the multiplicative inverse of in .Since in field all nonzero elements are invertible, it follows that the only proper ideal of is the zero ideal...

For some authors (e.g., Bourbaki, 1964), the same as principal ideal domain. Most authors, however, do not require the ring to be an integral domain, and define a principal ring (sometimes also called a principal ideal ring) simply as a commutative unit ring (different from the zero ring) in which every ideal is principal, i.e., can be generated by a single element. Examples include the ring of integers , any field, and any polynomial ring in one variable over a field. While all Euclidean rings are principal rings, the converse is not true.If the ideal of the commutative unit ring is generated by the element of , in any quotient ring the corresponding ideal is generated by the residue class of . Hence, every quotient ring of a principal ideal ring is a principal ideal ring as well. Since is a principal ideal domain, it follows that the rings are all principal ideal rings, though not all of them are principal ideal domains.Principal ideal rings which are..

A principal ideal domain is an integral domain in which every proper ideal can be generated by a single element. The term "principal ideal domain" is often abbreviated P.I.D. Examples of P.I.D.s include the integers, the Gaussian integers, and the set of polynomials in one variable with real coefficients.Every Euclidean ring is a principal ideal domain, but the converse is not true. Nevertheless, the notion of greatest common divisor arising from the Euclidean algorithm can be extended to the more general context of principal ideal domains as follows. Given two nonzero elements of a principal ideal domain , a greatest common divisor of and is defined as any element of such thatEvery principal ideal domain is a unique factorization domain, but not conversely. Every polynomial ring over a field is a unique factorization domain, but it is a principal ideal domain iff the number of indeterminates is one...

A von Neumann regular ring is a ring such that for all , there exists a satisfying (Jacobson 1989, p. 196).More formally, a ring is regular in the sense of von Neumann iff the following equivalent conditions hold. 1. Every -module is flat. 2. is a projective -module for every finitely generated ideal . 3. Every finitely generated right ideal is generatedby an idempotent. 4. Every finitely generated right ideal is a direct summand of .

A nonzero and noninvertible element of a ring which generates a prime ideal. It can also be characterized by the condition that whenever divides a product in , divides one of the factors. The prime elements of are the prime numbers .In an integral domain, every prime element is irreducible, but the converse holds only in unique factorization domains. The ring , where i is the imaginary unit, is not a unique factorization domain, and there the element 2 is irreducible, but not prime, since 2 divides the product , but it does not divide any of the factors.

A unique factorization domain, called UFD for short, is any integral domain in which every nonzero noninvertible element has a unique factorization, i.e., an essentially unique decomposition as the product of prime elements or irreducible elements. In this context, the two notions coincide, since in a unique factorization domain, every irreducible element is prime, whereas the opposite implication is true in every domain.This definition arises as an application of the fundamental theorem of arithmetic, which is true in the ring of integers , to more abstract rings. Other examples of unique factorization domains are the polynomial ring , where is a field, and the ring of Gaussian integers . In general, every principal ideal domain is a unique factorization domain, but the converse is not true, since every polynomial ring is a unique factorization domain, but it is not a principal ideal domain if ...

In an integral domain , the decomposition of a nonzero noninvertible element as a product of prime (or irreducible) factors(1)is unique if every other decomposition of the same type has the same number of factors(2)and its factors can be rearranged in such a way that for all indices , and differ by an invertible factor.The prime factorization of an element, if it exists, is always unique, but this does not apply, in general, to irreducible factorizations: in the ring ,(3)are two different irreducible factorizations, none of which is prime. 2 is not a prime element in , since it does not divide either of the factors of the middle expression. In fact(4)lie both outside . Furthermore,(5)which shows that is not prime either.An integral domain where every nonzero noninvertible element admits a unique irreducible factorization is called a unique factorization domain...

A ring defined on a singleton set . The ring operations (multiplication and addition) are defined in the only possible way,(1)and(2)It follows that this is a commutative unit ring, where is the multiplicative identity. Of course, also coincides with the additive identity, i.e., it is the so-called zero element of the ring. For this reason, the trivial ring is often denoted and also called the zero ring. In fact, the subset is the only trivial subring of the ring of integers .A unit ring is trivial whenever , since this equality implies that for all (3)A trivial ring is a trivial module over itself.

The ring of fractions of an integral domain. The field of fractions of the ring of integers is the rational field , and the field of fractions of the polynomial ring over a field is the field of rational functionsThe field of fractions of an integral domain is the smallest field containing , since it is obtained from by adding the least needed to make a field, namely the possibility of dividing by any nonzero element.

A extension ring (or ring extension) of a ring is any ring of which is a subring. For example, the field of rational numbers and the ring of Gaussian integers are extension rings of the ring of integers .For every ring , the polynomial ring is a ring extension of . If is a ring extension of , and , the setis the smallest subring of containing and , and is a ring extension of . More generally, given finitely many elements of , we can considerwhich is the ring extension of in generated by .

A proper ideal of a ring is called semiprime if, whenever for an ideal of and some positive integer, then . In other words, the quotient ring is a semiprime ring.If is a commutative ring, this is equivalent to requiring that coincides with its radical (and in this case is also called an ideal radical). This means that, whenever a certain positive integer power of an element of belongs to , the element itself lies in . A prime ideal is certainly semiprime, but the latter is a strictly more general notion. The ideal of the ring of integers is not prime, but it is semiprime, since for all integers , is a multiple of iff is, since both 2 and 3 must appear in its prime factorization. The same argument shows that the ideal of is always semiprime if is squarefree. This is not necessarily the case when is a semiprime number, which causes a conflict in terminology.In general, the semiprime ideals of a principal ideal domain are the proper ideals whose generator has no multiple..

A commutative Noetherian unit ring having only finitely many maximal ideals. A ring having the same properties except Noetherianity is called quasilocal.If is a field, the maximal ideals of the ring of polynomials in the indeterminate are the principal idealswhere is any element of . There is a one-to-one correspondence between these ideals and the elements of . Hence is semilocal if and only if is finite.A semilocal ring always has finite Krull dimension.The ring of integers is an example of a Noetherian nonsemilocal ring, since its maximal ideals are the principal ideals , where is any prime number.

Eisenstein's irreducibility criterion is a sufficient condition assuring that an integer polynomial is irreducible in the polynomial ring .The polynomialwhere for all and (which means that the degree of is ) is irreducible if some prime number divides all coefficients , ..., , but not the leading coefficient and, moreover, does not divide the constant term .This is only a sufficient, and by no means a necessary condition. For example, the polynomial is irreducible, but does not fulfil the above property, since no prime number divides 1. However, substituting for produces the polynomial , which does fulfill the Eisenstein criterion (with ) and shows the polynomial is irreducible.

In a noncommutative ring , a left ideal is a subset which is an additive subgroup of and such that for all and all ,A left ideal of can be characterized as a right ideal of the opposite ring of .In a commutative ring, the notions of right idealand left ideal coincide.

The most general form of this theorem states that in a commutative unit ring , the height of every proper ideal generated by elements is at most . Equality is attained if these elements form a regular sequence.Setting yields part of the original statement on principal ideals, also known under the German name Hauptidealsatz, that for every nonzero, noninvertible element of , the ideal of has height at most 1, and, moreover, iff is a non-zero divisor.It immediately follows as a corollary that every proper ideal of a Noetherian ring has finite height and that a principal ideal domain has Krull dimension equal to 1.

The extension ring obtained from a commutative unit ring (other than the trivial ring) when allowing division by all non-zero divisors. The ring of fractions of an integral domain is always a field.The term "ring of fractions" is sometimes used to denote any localization of a ring. The ring of fractions in the above meaning is then referred to as the total ring of fractions, and coincides with the localization with respect to the set of all non-zero divisors.When defining addition and multiplication of fractions, all that is required of the denominators is that they be multiplicatively closed, i.e., if , then ,(1)(2)Given a multiplicatively closed set in a ring , the ring of fractions is all elements of the form with and . Of course, it is required that and that fractions of the form and be considered equivalent. With the above definitions of addition and multiplication, this set forms a ring.The original ring may not embed in this ring of..

A ring in which the zero ideal is an irreducible ideal. Every integral domain is irreducible since if and are two nonzero ideals of , and , are nonzero elements, then is a nonzero element of , which therefore cannot be the zero ideal.

A proper ideal of a ring that is not the intersection of two ideals which properly contain it. In a principal ideal domain, the ideal is irreducible iff or is an irreducible element.

An element of an extension field of a field which is not algebraic over . A transcendental number is a complex number which is transcendental over the field of rational numbers.

An improper use of the symbol for the imaginary unit leads to the apparent proof of a false statement.(1)(2)(3)(4)(5)The reason for the fallacy is that is not an ordinary (real) square root, hence the rule for computing the quotient of radicals does not apply to it.

An optical illusion named after British psychologist James Fraser, who first studied the illusion in 1908 (Fraser 1908). The illusion is also known as the false spiral, or by its original name, the twisted cord illusion. While the image appears to be a spiral formed by a rope containing twisted strands of two different colors, it actually consists of concentric circles of twisted cords.The visual distortion is produced by combining a regular line pattern (the circles) with misaligned parts (the differently colored strands). Zöllner's illusion and the café wall illusion are based on a similar principle, like many other visual effects, in which a sequence of tilted elements causes the eye to perceive phantom twists and deviations.

The café wall illusion, sometimes also called the Münsterberg illusion (Ashton Raggatt McDougall 2006), is an optical illusion produced by a black and white rectangular tessellation when the tiles are shifted in a zigzag pattern, as illustrated above. While the pattern seems to diverge towards the upper and lower right corners in the upper figure, the gray lines are actually parallel. Interestingly, the illusion greatly diminishes if black lines are used instead of gray.Gregory and Heard (1979) first noticed the illusion on the wall decoration of a café in Bristol, England. The café wall illusion is only one among many visual distortion effects involving parallel lines. The most famous example of this kind is Zöllner's illusion.The image above shows a picture of a building in Melbourne, Australia designed to exhibit this illusion (C. L. Taylor, pers. comm., Aug. 5, 2006). The building,..

Given metric spaces , with metrics respectively, the product metric is a metric on the Cartesian product defined asThis definition can be extended to the product of countably many metric spaces.If for all , and is the Euclidean metric of the real line, the product metric induces the Euclidean topology of the -dimensional Euclidean space . It does not coincide with the Euclidean metric of , but it is equivalent to it.

The taxicab metric, also called the Manhattan distance, is the metricof the Euclidean plane defined byfor all points and . This number is equal to the length of all paths connecting and along horizontal and vertical segments, without ever going back, like those described by a car moving in a lattice-like street pattern.

A topology that is "potentially" a metric topology, in the sense that one can define a suitable metric that induces it. The word "potentially" here means that although the metric exists, it may be unknown.In fact, there are sufficient criteria on the topology that assure the existence of such a metric even if this is not explicitly given. An example of an existence theorem of this kind is due to Urysohn (Kelley 1955, p. 125), who proved that a regular T1-space whose topology has a countable basis is metrizable.Conversely, a metrizable space is always and regular, but the condition on the basis has to be weakened since in general, it is only true that the topology has a basis which is formed by countably many locally finite families of open sets.Special metrizability criteria are known for T2-spaces. A compact -space is metrizable iff the set of all elements of is a zero set (Willard 1970, p. 163). The continuous image..

A topology induced by the metric defined on a metric space . The open sets are all subsets that can be realized as the unions of open ballswhere , and .The metric topology makes a T2-space. Given two distinct points and of , their distance is certainly positive, so the open balls and are disjoint neighborhoods of and , respectively.

A metric on a bunch of segments with a common endpoint , which defines the distance between two points and as the length of the shortest path connecting them inside this geometric configuration. If and lie on the same segment, this is the length of segment . Otherwise, it is the sum of the lengths of segment and segment .

Two metrics and defined on a space are called equivalent if they induce the same metric topology on . This is the case iff, for every point of , every ball with center at defined with respect to :(1)contains a ball with center with respect to :(2)and conversely.Every metric on has uncountably many equivalent metrics. For every positive real number , a "scaled" metric can be defined such that for all ,(3)In fact, for all :(4)Another metric equivalent to is defined by(5)for all . In fact,(6)and(7)In the Euclidean plane , the metric(8)with circular balls can be defined in addition to the Euclidean metric. An equivalent more general metric for all positive real numbers and can be defined as(9)with elliptic balls, and the taxicab metric(10)can be defined with square "balls." All these are equivalent to the Euclidean metric...

The metric defined on a nonempty set by(1)(2)if for all .It follows that the open ball of radius and center at (3)is(4)The metric topology induced is the discrete topology.

A distance on a set that fulfils the same properties as a metric except relaxes the definition to allow the distance between two different points to be zero.An example of pseudometric on the set of all functions is defined by . It is nonnegative, symmetric, fulfils the triangle inequality and the condition , but it is also true that .

A nonnegative function describing the "distance" between neighboring points for a given set. A metric satisfies the triangle inequality(1)and is symmetric, so(2)A metric also satisfies(3)as well as the condition that implies . If this latter condition is dropped, then is called a pseudometric instead of a metric.A set possessing a metric is called a metric space. When viewed as a tensor, the metric is called a metric tensor.

Codimension is a term used in a number of algebraic and geometric contexts to indicate the difference between the dimension of certain objects and the dimension of a smaller object contained in it. This rough definition applies to vector spaces (the codimension of the subspace in is ) and to topological spaces (with respect to the Euclidean topology and the Zariski topology, the codimension of a sphere in is ).The first example is a particular case of the formula(1)which gives the codimension of a subspace of a finite-dimensional abstract vector space . The second example has an algebraic counterpart in ring theory. A sphere in the three-dimensional real Euclidean space is defined by the following equation in Cartesian coordinates(2)where the point is the center and is the radius. The Krull dimension of the polynomial ring is 3, the Krull dimension of the quotient ring(3)is 2, and the difference is also called the codimension of the ideal(4)According..

A set in which can be reduced to one of its points, say , by a continuous deformation, is said to be contractible. The transformation is such that each point of the set is driven to through a path with the properties that 1. Each path runs entirely inside the set. 2. Nearby points move on "neighboring" paths. Condition (1) implies that a disconnected set,i.e., a set consisting of separate parts, cannot be contractible.Condition (2) implies that the circumference of a circle is not contractible. The latter follows by considering two near points and lying on different sides of a point . The paths connecting and with are either opposite each other or have different lengths. A similar argument shows that, in general, for all , the -sphere (i.e., the boundary of the -dimensional ball) is not contractible.A gap or a hole in a set can be an obstruction to contractibility. There are, however, examples of contractible sets with holes, for example,..

The homomorphism which, according to the snake lemma, permits construction of an exact sequence(1)from the above commutative diagram with exact rows. The homomorphism is defined by(2)for all , denotes the image, and is obtained through the following construction, based on diagram chasing.1. Exploit the surjectivity of to find such that . 2. Since because of the commutativity of the right square, belongs to , which is equal to due to the exactness of the lower row at . This allows us to find such that . While the elements and are not uniquely determined, the coset is, as can be proven by using more diagram chasing. In particular, if and are other elements fulfilling the requirements of steps (1) and (2), then and , and(3)hence because of the exactness of the upper row at . Let be such that(4)Then(5)because the left square is commutative. Since is injective, it follows that(6)and so(7)..

A commutative diagram is a collection of maps in which all map compositions starting from the same set and ending with the same set give the same result. In symbols this means that, whenever one can form two sequences(1)and(2)the following equality holds:(3)Commutative diagrams are usually composed by commutative triangles and commutative squares.Commutative triangles and squares can also be combined to form plane figures or space arrangements.A commutative diagram can also contain multiple arrows that indicate different maps between the same two sets.A looped arrow indicates a map from a set to itself.The above commutative diagram expresses the fact that is the inverse map to , since it is a pictorial translation of the map equalities and .This can also be represented using two separate diagrams.Many other mathematical concepts and properties, especially in algebraic topology, homological algebra, and category theory, can be formulated..

Betti numbers are topological objects which were proved to be invariants by Poincaré, and used by him to extend the polyhedral formula to higher dimensional spaces. Informally, the Betti number is the maximum number of cuts that can be made without dividing a surface into two separate pieces (Gardner 1984, pp. 9-10). Formally, the th Betti number is the rank of the th homology group of a topological space. The following table gives the Betti number of some common surfaces.surfaceBetti numbercross-cap1cylinder1klein bottle2Möbius strip1plane lamina0projective plane1sphere0torus2Let be the group rank of the homology group of a topological space . For a closed, orientable surface of genus , the Betti numbers are , , and . For a nonorientable surface with cross-caps, the Betti numbers are , , and .The Betti number of a finitely generated Abelian group is the (uniquely determined) number such thatwhere , ..., are finite cyclic..

A general class of means introduced by Italian mathematicianOscar Chisini (pronounced keeseenee) in 1929.Given a function of variables , the Chisini mean of values associated with is defined as the number such thatOf course, the function must be chosen in such a way that there always is exactly one number with this property.The most common means are Chisini means associated with the functions listed in the following table. Every weighted mean corresponds to the weight vector .meanfunction arithmetic meanweighted arithmetic meangeometric meanweighted geometric meanharmonic meanweighted harmonic meanquadratic meanweighted quadratic mean

The group direct sum of a sequence of groups is the set of all sequences , where each is an element of , and is equal to the identity element of for all but a finite set of indices . It is denoted(1)and it is a group with respect to the componentwise operation derived from the operations of the groups .This definition can easily be extended to any collection of groups, where is any finite or infinite set of indices.If the additional condition on the identity elements is dropped, we get the definition of the group direct product. Hence, the two notions coincide whenever the set of indices is finite. Thus, for any groups and ,(2)denote the same object.If and are subgroups of the same additive group , the equality(3)conventionally means that every has a unique decomposition , where and , so that is essentially the same as the set of all ordered pairs ...

Given the direct sum of additive Abelian groups , and are called direct summands. The map defined by is called the injection of the first summand, and the map defined by is called the projection onto the first summand. Similar maps are defined for the second summand .The above definitions extend in a natural way to the direct sums of more than twoAbelian groups.

Given two normal subgroups and of a group, and two normal subgroups and of and respectively,(1)(2)and one has an isomorphism of quotient groups(3)(Zassenhaus 1934). This lemma was named by Serge Lang (2002, pp. 20-21) based on the shape of the diagram above, which Lang derived from Zassenhaus's original publication.The butterfly lemma visualizes the inclusion between subgroups. In particular, whenever two groups are connected by a segment to a point lying right above, this point represents their product, and whenever the point lies right below, it represents their intersection. This diagram is part of the Hasse diagram of the partially ordered set of subgroups of the given group. The quotient groups formed along the three central vertical lines are all isomorphic.The butterfly lemma can be used to prove the equivalence of compositionseries in Jordan-Hölder theorem...

In general, groups are not Abelian. However, there is always a group homomorphism to an Abelian group, and this homomorphism is called Abelianization. The homomorphism is abstractly described by its kernel, the commutator subgroup , which is the unique smallest normal subgroup of such that the quotient group is Abelian. Roughly speaking, in any expression, every product becomes commutative after Abelianization. As a consequence, some previously unequal expressions may become equal, or even represent the identity element.For example, in the eight-element quaternion group , the commutator subgroup is . The Abelianization of is a copy of , and for instance, in the Abelianization.

A group automorphism is an isomorphism from a group to itself. If is a finite multiplicative group, an automorphism of can be described as a way of rewriting its multiplication table without altering its pattern of repeated elements. For example, the multiplication table of the group of 4th roots of unity can be written as shown above, which means that the map defined byis an automorphism of .The map is also a group automorphism for as long as is not congruent to 0. Conjugating by a fixed element is a group automorphism called an inner automorphism.In general, the automorphism group of an algebraic object , like a ring or field, is the set of isomorphisms of that object , and is denoted . It forms a group by composition of maps. For a fixed group , the collection of group automorphisms is the automorphism group ...

The kernel of a group homomorphism is the set of all elements of which are mapped to the identity element of . The kernel is a normal subgroup of , and always contains the identity element of . It is reduced to the identity element iff is injective.

A subset of an algebraic variety which is itself a variety. Every variety is a subvariety of itself; other subvarieties are called proper subvarieties.A sphere of the three-dimensional Euclidean space is an algebraic variety since it is defined by a polynomial equation. For example,(1)defines the sphere of radius 1 centered at the origin. Its intersection with the -plane is a circle given by the system of polynomial equations:(2)(3)Hence the circle is itself an algebraicvariety, and a subvariety of the sphere, and of the plane as well.Whenever some new independent equations are added to the equations defining a certain variety, the resulting variety will be smaller, since its points will be subject to more conditions than before. In the language of ring theory, this means that, while the sphere is the zero set of all polynomials of the ideal of , every subvariety of it will be defined by a larger ideal; this is the case for , which is the defining..

A class of subvarieties of the Grassmannian . Given integers , the Schubert variety is the set of points of representing the -dimensional subspaces of such that, for all ,It is a projective algebraic varietyof dimension

Given an affine variety in the -dimensional affine space , where is an algebraically closed field, the coordinate ring of is the quotient ringwhere is the ideal formed by all polynomials with coefficients in which are zero at all points of . If is the entire -dimensional affine space , then this ideal is the zero ideal. It follows that the coordinate ring of is the polynomial ring . The coordinate ring of a plane curve defined by the Cartesian equation in the affine plane is .In general, the Krull dimension of ring is equal to the dimension of as a closed set of the Zariski topology of .Two polynomials and define the same function on iff . Hence the elements of are equivalence classes which can be identified with the polynomial functions from to .

According to many authors (e.g., Kelley 1955, p. 112; Joshi 1983, p. 162; Willard 1970, p. 99) a normal space is a topological space in which for any two disjoint closed sets there are two disjoint open sets and such that and .Other authors (e.g., Cullen 1968, p. 118) define the notion differently, using separation axioms.

The subset of the Euclidean plane formed by the union of the x-axis, the line segment with interval of the y-axis, and the sequence of segments with endpoints and for all positive integers .With respect to the relative topology is pathwise-connected. It is therefore connected, but not locally pathwise-connected at any point of the open interval since each open disk centered at point one of these points intersects in a union of parallel segments, forming a disconnected set.

The vector space generated by the columns of a matrix viewed as vectors. The column space of an matrix with real entries is a subspace generated by elements of , hence its dimension is at most . It is equal to the dimension of the row space of and is called the rank of .The matrix is associated with a linear transformation , defined byfor all vectors of , which we suppose written as column vectors. Note that is the product of an and an matrix, hence it is an matrix according to the rules of matrix multiplication. In this framework, the column vectors of are the vectors , where are the elements of the standard basis of . This shows that the column space of is the range of , and explains why the dimension of the latter is equal to the rank of .

The subset of the Euclidean plane formed by the union of the interval of the x-axis and all line segments of unit length passing through the origin which form an angle (measured in radians) with it, for all positive integers .With respect to the relative topology, is pathwise-connected. Therefore it is connected, but it is not locally pathwise-connected at any point of the open interval . Each disk centered at one of these points intersects in a union of disjoint segments, which form a disconnected set.Let be the broom space formed by segments of length for all natural numbers , and place , , , ... one right after the other on the -axis. This will cover the half-open interval of the -axis (above figure). The space obtained by adding the point (2,0) to this sequence of brooms is then connected im kleinen at point (2,0), since each open neighborhood of (2,0) contains a closed disk whose radius is exactly formed by the basis intervals of for all sufficiently..

The vector space generated by the rows of a matrix viewed as vectors. The row space of a matrix with real entries is a subspace generated by elements of , hence its dimension is at most equal to . It is equal to the dimension of the column space of (as will be shown below), and is called the rank of .The row vectors of are the coefficients of the unknowns in the linear equation system(1)where(2)and is the zero vector in . Hence, the solutions span the orthogonal complement to the row space in , and(3)On the other hand, the space of solutions also coincides with the kernel (or null space) of the linear transformation , defined by(4)for all vectors of . And it also true that(5)where denotes the kernel and the image, since the nullity and the rank always add up to the dimension of the domain. It follows that the dimension of the row space is(6)which is equal to the dimension of the column space...

A metric topology induced by the Euclidean metric. In the Euclidean topology of the -dimensional space , the open sets are the unions of -balls. On the real line this means unions of open intervals. The Euclidean topology is also called usual or ordinary topology.

For every topological T1-space , the following conditions are equivalent. 1. is regular and second countable, 2. is separable and metrizable. 3. is homeomorphic to a subspace of the Hilbert cube.

A characterization of normal spaces which states that a topological space is normal iff, for any two nonempty closed disjoint subsets , and of , there is a continuous map such that and . A function with this property is called a Urysohn function.This formulation refers to the definition of normal space given by Kelley (1955, p. 112) or Willard (1970, p. 99). In the statement for an alternative definition (e.g., Cullen 1968, p. 118), the word "normal" has to be replaced by .

A product space is compact iff is compact for all . In other words, the topological product of any number of compact spaces is compact. In particular, compactness is a productive property. As a consequence, every Hilbert cube is compact.This statement implies the axiom of choice, asproven by Kelley (1950).

A diagram lemma which states that, given the above commutative diagram with exact rows, the following holds: 1. If is surjective, and and are injective, then is injective; 2. If is injective, and and are surjective, then is surjective. This lemma is closely related to the five lemma, whichis based on a similar diagram obtained by adding a single column.

A diagram lemma which states that, given the commutative diagram of additive Abelian groups with exact rows, the following holds: 1. If is surjective, and and are injective, then is injective; 2. If is injective, and and are surjective, then is surjective. If and are bijective, the hypotheses of (1) and (2) are satisfied simultaneously, and the conclusion is that is bijective. This statement is known as the Steenrod five lemma.If , , , and are the zero group, then and are zero maps, and thus are trivially injective and surjective. In this particular case the diagram reduces to that shown above. It follows from (1), respectively (2), that is injective (or surjective) if and are. This weaker statement is sometimes referred to as the "short five lemma."

A problem posed by the Slovak mathematician Stefan Znám in 1972 asking whether, for all integers , there exist integers all greater than 1 such that is a proper divisor of for each . The answer is negative for (Jának and Skula 1978) and affirmative for (Sun Qi 1983). Sun Qi also gave a lower bound for the number of solutions.All solutions for have now been computed, summarized in the table below. The numbers of solutions for , 3, ... terms are 0, 0, 0, 2, 5, 15, 93, ... (OEIS A075441), and the solutions themselves are given by OEIS A075461.known solutions references20--Jának and Skula (1978)30--Jának and Skula (1978)40--Jának and Skula (1978)522, 3, 7, 47, 3952, 3, 11, 23, 31652, 3, 7, 43, 1823, 1936672, 3, 7, 47, 403, 194032, 3, 7, 47, 415, 81112, 3, 7, 47, 583, 12232, 3, 7, 55, 179, 243237152, 3, 7, 43, 1807, 3263447, 2130014000915Jának and Skula (1978)2, 3, 7, 43, 1807, 3263591, 71480133827Cao, Liu, and Zhang..

According to Pólya, the Cartesian pattern is the resolution method for arithmetical or geometrical problems based on equations. The first step is to translate the question into one or more algebraic equalities, which express the relationship between the numerical data (the coefficients) and the quantities to be determined (the unknowns). This relationship can be described in text, or be depicted in a figure.The second step is to solve the equations.Normally, the quantity requested by the problem is only one, which permits us to reduce the procedure to a single equation, whose sides contain two different expressions of the same quantity. Consider, for example, a problem asking for one of the legs of a right triangle given that the length of this leg is half the length of the hypotenuse and that the other leg has length 1. If the unknown leg is denoted by and the hypotenuse by , then(1)as specified, and, moreover,(2)by the Pythagorean theorem...

A collection of open sets of a topological space whose union contains a given subset. For example, an open cover of the real line, with respect to the Euclidean topology, is the set of all open intervals , where .The set of all intervals , where , is an open cover of the open interval .

"Neighborhood" is a word with many different levels of meaning in mathematics.One of the most general concepts of a neighborhood of a point (also called an epsilon-neighborhood or infinitesimal open set) is the set of points inside an -ball with center and radius . A set containing an open neighborhood is also called a neighborhood.The graph neighborhood of a vertex in a graph is the set of all the vertices adjacent to generally including itself. More generally, the th neighborhood of is the set of all vertices that lie at the distance from . The subgraph induced by the neighborhood of a graph from vertex (again, most commonly including itself) is called the neighborhood graph (or sometimes "ego graph" in more recent literature).

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