|
Minkowski's Theorem
In mathematics, Minkowski's theorem is the statement that every convex set in \mathbb^n which is symmetric with respect to the origin and which has volume greater than 2^n contains a non-zero integer point (meaning a point in \Z^n that is not the origin). The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers. It can be extended from the integers to any lattice L and to any symmetric convex set with volume greater than 2^n\,d(L), where d(L) denotes the covolume of the lattice (the absolute value of the determinant of any of its bases). Formulation Suppose that is a lattice of determinant in the - dimensional real vector space and is a convex subset of that is symmetric with respect to the origin, meaning that if is in then is also in . Minkowski's theorem states that if the volume of is strictly greater than , then must contain at least one lattice point other than the origin. (Sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Lattice
In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid lattice. is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice. Automorphism group The automorphism group (or group of congruences) of the integer lattice consists of all permutations and sign changes of the coordinates, and is of order 2''n'' ''n''!. As a matrix group it is given by the set of all ''n''×''n'' signed permutation matrices. This group is isomorphic to the semidirect product :(\mathbb Z_2)^n \rtimes S_n where the symmetric group ''S''''n'' acts on (Z2)''n'' by permutation (this is a classic example of a wreath product). For the square lattice, this is the group of the square, or the dihedral group of order 8; for the three-dimensional cubic lattice, we get the group of the cube, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Problem
In computer science, lattice problems are a class of Mathematical optimization, optimization problems related to mathematical objects called Lattice (group), lattices. The conjectured intractability of such problems is central to the construction of secure Lattice-based cryptography, lattice-based cryptosystems: Lattice problems are an example of NP-hardness, NP-hard problems which have been shown to be Computational hardness assumption, average-case hard, providing a test case for the security of cryptographic algorithms. In addition, some lattice problems which are worst-case hard can be used as a basis for extremely secure cryptographic schemes. The use of worst-case hardness in such schemes makes them among the very few schemes that are very likely secure even against quantum computers. For applications in such cryptosystems, lattices over vector space (often \mathbb^n) or free modules (often \mathbb^n) are generally considered. For all the problems below, assume that we are g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Constant
In mathematics, the Hermite constant, named after Charles Hermite, determines how short an element of a lattice in Euclidean space can be. The constant ''γn'' for integers ''n'' > 0 is defined as follows. For a lattice ''L'' in Euclidean space R''n'' with unit covolume, i.e. vol(R''n''/''L'') = 1, let ''λ''1(''L'') denote the least length of a nonzero element of ''L''. Then is the maximum of ''λ''1(''L'') over all such lattices ''L''. The square root in the definition of the Hermite constant is a matter of historical convention. Alternatively, the Hermite constant ''γn'' can be defined as the square of the maximal systole of a flat ''n''-dimensional torus of unit volume. Example The Hermite constant is known in dimensions 1–8 and 24. For ''n'' = 2, one has ''γ''2 = . This value is attained by the hexagonal lattice of the Eisenstein integers. Estimates It is known thatKitaoka (1993) p. 36 :\gamma_n \le \left( \frac 4 3 \right)^\frac. A stronger estim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) ''endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blichfeldt's Theorem
Blichfeldt's theorem is a mathematical theorem in the geometry of numbers, stating that whenever a bounded set in the Euclidean plane has area A, it can be translated so that it includes at least \lceil A\rceil points of the integer lattice. Equivalently, every bounded set of area A contains a set of \lceil A\rceil points whose coordinates all differ by integers. This theorem can be generalized to other lattices and to higher dimensions, and can be interpreted as a continuous version of the pigeonhole principle. It is named after Danish-American mathematician Hans Frederick Blichfeldt, who published it in 1914. Some sources call it Blichfeldt's principle or Blichfeldt's lemma. Statement and proof The theorem can be stated most simply for points in the Euclidean plane, and for the integer lattice in the plane. For this version of the theorem, let S be any measurable set, let A denote its area, and round this number up to the next integer value, n=\lceil A\rceil . Then Blichfeldt's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositive statement.) In other words, every element of the function's codomain is the image of one element of its domain. The term must not be confused with that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an is also called a . However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see for more details. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proof By Contradiction
In logic and mathematics, proof by contradiction is a form of proof that establishes the truth or the validity of a proposition, by showing that assuming the proposition to be false leads to a contradiction. Proof by contradiction is also known as indirect proof, proof by assuming the opposite, and ''reductio ad impossibile''. It is an example of the weaker logical refutation ''reductio ad absurdum''. A mathematical proof employing proof by contradiction usually proceeds as follows: #The proposition to be proved is ''P''. #We assume ''P'' to be false, i.e., we assume ''¬P''. #It is then shown that ''¬P'' implies falsehood. This is typically accomplished by deriving two mutually contradictory assertions, ''Q'' and ''¬Q'', and appealing to the Law of noncontradiction. #Since assuming ''P'' to be false leads to a contradiction, it is concluded that ''P'' is in fact true. An important special case is the existence proof by contradiction: in order to demonstrate the existence of an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypercube
In geometry, a hypercube is an ''n''-dimensional analogue of a square () and a cube (). It is a closed, compact, convex figure whose 1- skeleton consists of groups of opposite parallel line segments aligned in each of the space's dimensions, perpendicular to each other and of the same length. A unit hypercube's longest diagonal in ''n'' dimensions is equal to \sqrt. An ''n''-dimensional hypercube is more commonly referred to as an ''n''-cube or sometimes as an ''n''-dimensional cube. The term measure polytope (originally from Elte, 1912) is also used, notably in the work of H. S. M. Coxeter who also labels the hypercubes the γn polytopes. The hypercube is the special case of a hyperrectangle (also called an ''n-orthotope''). A ''unit hypercube'' is a hypercube whose side has length one unit. Often, the hypercube whose corners (or ''vertices'') are the 2''n'' points in R''n'' with each coordinate equal to 0 or 1 is called ''the'' unit hypercube. Construction A hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Jargon
The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense. Some phrases, like "in general", appear below in more than one section. Philosophy of mathematics ; abstract nonsense:A tongue-in-cheek reference to category theory, using which one can employ arguments that establish a (possibly concrete) result without reference to any specifics of the present problem. For that reason, it's also known as ''general abstract nonsense'' or ''generalized abstract nonsense''. ; canonical:A reference to a standard or choice-free presentation of some mathematical object (e.g., canonical map, canonical form, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Area
Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an object or its external boundary, outline, or external Surface (mathematics), surface, as opposed to other properties such as color, Surface texture, texture, or material type. A pl ... or planar lamina, while ''surface area'' refers to the area of an open surface or the boundary (mathematics), boundary of a solid geometry, three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analogue of the length of a plane curve, curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). The area of a shape can be measured by com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
