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Congruence
Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In modular arithmetic, having the same remainder when divided by a specified integer **Ramanujan's congruences, congruences for the partition function, , first discovered by Ramanujan in 1919 **Congruence subgroup, a subgroup defined by congruence conditions on the entries of a matrix group with integer entries **Congruence of squares, in number theory, a congruence commonly used in integer factorization algorithms * Matrix congruence, an equivalence relation between two matrices * Congruence (manifolds), in the theory of smooth manifolds, the set of integral curves defined by a nonvanishing vector field defined on the manifold * Congruence (general relativity), in general relativity, a congruence in a four-dimensional Lorentzian manifold that is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence Subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible matrix, invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are ''even''. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer. The existence of congruence subgroups in an arithmetic group provides it with a wealth of subgroups, in particular it shows that the group is residually finite. An important question regarding the algebraic structure of arithmetic groups is the congruence subgroup problem, which asks whether all subgroups of finite Index of a subgroup, index are essentially congruence subgroups. Congruence subgroups of 2×2 matrices are fundamental objects in the classical the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence Relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation. Basic example The prototypical example of a congruence relation is congruence modulo n on the set of integers. For a given positive integer n, two integers a and b are called congruent modulo n, written : a \equiv b \pmod if a - b is divisible by n (or equivalently if a and b have the same remainder when divided by n). For example, 37 and 57 are congruent modulo 10, : 37 \equiv 57 \pmod since 37 - 57 = -20 is a multiple of 10, or equivalently since both 37 and 57 have a remainder of 7 when divided by 10. Congruence modulo n (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence (general Relativity)
In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation. Types of congruences Congruences generated by nowhere vanishing timelike, null, or spacelike vector fields are called ''timelike'', ''null'', or ''spacelike'' respectively. A congruence is called a ''geodesic congruence'' if it admits a tangent vector field \vec with vanishing covariant derivative, \nabla_ \vec = 0. Relation with vector fields The integral curves of the vector field are a family of ''non-intersecting'' parameterized curves which fill up the spacetime. The congruence consists of the curves themselves, without reference to a particular parameterization. Many distinct vector fields can give rise to the ''same'' congruen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called congruent if, and only if, one can be transformed into the other by an isometry, i.e., a combination of rigid motions, namely a translation, a rotation, and a reflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. Therefore two distinct plane figures on a piece of paper are congruent if they can be cut out and then matched up completely. Turning the paper over is permitted. In elementary geometry the word ''congruent'' is often used as follows. The word ''equal'' is often used in place of ''congruent'' for these objects. *Two line segments are congruent if they have the same length. *Two angles are congruent if they have the same measure. *Two circles are congruent if they have the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan's Congruences
In mathematics, Ramanujan's congruences are some remarkable congruences for the partition function ''p''(''n''). The mathematician Srinivasa Ramanujan discovered the congruences : \begin p(5k+4) & \equiv 0 \pmod 5, \\ p(7k+5) & \equiv 0 \pmod 7, \\ p(11k+6) & \equiv 0 \pmod . \end This means that: * If a number is 4 more than a multiple of 5, i.e. it is in the sequence :: 4, 9, 14, 19, 24, 29, . . . : then the number of its partitions is a multiple of 5. * If a number is 5 more than a multiple of 7, i.e. it is in the sequence :: 5, 12, 19, 26, 33, 40, . . . : then the number of its partitions is a multiple of 7. * If a number is 6 more than a multiple of 11, i.e. it is in the sequence :: 6, 17, 28, 39, 50, 61, . . . : then the number of its partitions is a multiple of 11. Background In his 1919 paper, he proved the first two congruences using the following identities (using q-Pochhammer symbol notation): : \begin & \sum_^\infty p(5k+4)q^k=5\frac, \\ pt& \sum_^\infty p(7k+ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mood Congruence
Mood congruence is the consistency between a person's emotional state with the broader situations and circumstances being experienced by the persons at that time. By contrast, mood incongruence occurs when the individual's reactions or emotional state appear to be in conflict with the situation. In the context of psychosis, hallucinations and delusions may be considered mood congruent (such as feelings of personal inadequacy, guilt, or worthlessness during a bipolar disorder depressive episode) or incongruent. Background and theorists An important consideration to the difference between mood congruence and mood dependent (or state-dependent) memory is the determination that one cannot make accurate assumptions about the emotional state of a memory during the encoding process. Therefore, the memory that is recalled is not dependent on the affective state during encoding. Another important difference is that there are multiple memories that can be recalled while in particular mood s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence (manifolds)
In the theory of smooth manifolds, a congruence is the set of integral curves defined by a nonvanishing vector field defined on the manifold. Congruences are an important concept in general relativity, and are also important in parts of Riemannian geometry. A motivational example The idea of a congruence is probably better explained by giving an example than by a definition. Consider the smooth manifold R². Vector fields can be specified as ''first order linear partial differential operators'', such as :\vec = ( x^2 - y^2 ) \, \partial_x + 2 \, x y \, \partial_y These correspond to a system of ''first order linear ordinary differential equations'', in this case :\dot = x^2 - y^2,\; \dot = 2 \, x y where dot denotes a derivative with respect to some (dummy) parameter. The solutions of such systems are ''families of parameterized curves'', in this case : x(\lambda) = \frac : y(\lambda) = \frac This family is what is often called a ''congruence of curves'', or just ''congrue ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence Of Squares
In number theory, a congruence of squares is a congruence commonly used in integer factorization algorithms. Derivation Given a positive integer ''n'', Fermat's factorization method relies on finding numbers ''x'' and ''y'' satisfying the equality :x^2 - y^2 = n We can then factor ''n'' = ''x''2 − ''y''2 = (''x'' + ''y'')(''x'' − ''y''). This algorithm is slow in practice because we need to search many such numbers, and only a few satisfy the equation. However, ''n'' may also be factored if we can satisfy the weaker congruence of squares condition: :x^2 \equiv y^2 \pmod :x \not\equiv \pm y \,\pmod From here we easily deduce :x^2 - y^2 \equiv 0 \pmod :(x + y)(x - y) \equiv 0 \pmod This means that ''n'' divides the product (''x'' + ''y'')(''x'' − ''y''). Thus (''x'' + ''y'') and (''x'' − ''y'') each contain factors of ''n'', but those factors can be trivial. In this case we need to find another ''x'' and ''y''. Computing the greatest common divisors of (''x'' +&t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zeller's Congruence
Zeller's congruence is an algorithm devised by Christian Zeller in the 19th century to calculate the day of the week for any Julian calendar, Julian or Gregorian calendar date. It can be considered to be based on the conversion between Julian day and the calendar date. Formula For the Gregorian calendar, Zeller's congruence is :h = \left(q + \left\lfloor\frac\right\rfloor + K + \left\lfloor\frac\right\rfloor + \left\lfloor\frac\right\rfloor - 2J\right) \bmod 7, for the Julian calendar it is :h = \left(q + \left\lfloor\frac\right\rfloor + K + \left\lfloor\frac\right\rfloor + 5 - J\right) \bmod 7, where * ''h'' is the day of the week (0 = Saturday, 1 = Sunday, 2 = Monday, ..., 6 = Friday) * ''q'' is the day of the month * ''m'' is the month (3 = March, 4 = April, 5 = May, ..., 14 = February) * ''K'' the year of the century (year \bmod 100). * ''J'' is the zero-based numbering, zero-based century (actually \lfloor year/100 \rfloor) For example, the zero-based centuries for 1995 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence Bias
Congruence bias is the tendency of people to over-rely on testing their initial hypothesis (the most ''congruent'' one) while neglecting to test alternative hypotheses. That is, people rarely try experiments that could disprove their initial belief, but rather try to repeat their initial results. It is a special case of the confirmation bias. Examples Suppose that, in an experimental setting, a subject is presented with two buttons and told that pressing one of those buttons, but not the other, will open a door. The subject adopts the hypothesis that the button on the left opens the door in question. A direct test of this hypothesis would be pressing the button on the left; an indirect test would be pressing the button on the right. The latter is still a valid test because once the result of the door's remaining closed is found, the left button is proven to be the desired button. (This example is parallel to Bruner, Goodnow, and Austin's example in the psychology classic, '' A Stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Congruence
In mathematics, two square matrices ''A'' and ''B'' over a field are called congruent if there exists an invertible matrix ''P'' over the same field such that :''P''T''AP'' = ''B'' where "T" denotes the matrix transpose. Matrix congruence is an equivalence relation. Matrix congruence arises when considering the effect of change of basis on the Gram matrix attached to a bilinear form or quadratic form on a finite-dimensional vector space: two matrices are congruent if and only if they represent the same bilinear form with respect to different bases. Note that Halmos defines congruence in terms of conjugate transpose (with respect to a complex inner product space) rather than transpose, but this definition has not been adopted by most other authors. Congruence over the reals Sylvester's law of inertia states that two congruent symmetric matrices with real entries have the same numbers of positive, negative, and zero eigenvalues. That is, the number of eigenvalues of ea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
