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Gauss Circle Problem
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and with radius r. This number is approximated by the area of the circle, so the real problem is to accurately bound the error term describing how the number of points differs from the area. The first progress on a solution was made by Carl Friedrich Gauss, hence its name. The problem Consider a circle in \mathbb^2 with center at the origin and radius r\ge 0. Gauss's circle problem asks how many points there are inside this circle of the form (m,n) where m and n are both integers. Since the equation of this circle is given in Cartesian coordinates by x^2+y^2= r^2, the question is equivalently asking how many pairs of integers ''m'' and ''n'' there are such that :m^2+n^2\leq r^2. If the answer for a given r is denoted by N(r) then the following list shows the first few values of N(r) for ''r'' an integer between 0 and 12 followed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grid Points In Radius-5 Circle
Grid, The Grid, or GRID may refer to: Common usage * Cattle grid or stock grid, a type of obstacle is used to prevent livestock from crossing the road * Grid reference, used to define a location on a map Arts, entertainment, and media * News grid, used in communications/public relations Fictional entities * Grid (comics), a fictional character in the DC Comics Universe * Grid (Jotun), Gríðr, a giantess in Norse mythology * The grid, the virtual environment of the game ''Second Life ''Second Life'' is an online multimedia platform that allows people to create an avatar for themselves and then interact with other users and user created content within a multi player online virtual world. Developed and owned by the San Fra ...'' * ''The Grid'', the computerized virtual world in which the Tron (franchise), Tron franchise exists Games and gaming * Nvidia GRID, a cloud gaming platform for Nvidia Tegra products * ''Power Grid'', the English-language edition of the multiplayer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Functions
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of ''n''". An example of an arithmetic function is the divisor function whose value at a positive integer ''n'' is equal to the number of divisors of ''n''. There is a larger class of number-theoretic functions that do not fit the above definition, for example, the prime-counting functions. This article provides links to functions of both classes. Arithmetic functions are often extremely irregular (see table), but some of them have series expansions in terms of Ramanujan's sum. Multiplicative and additive functions An arithmetic function ''a'' is * completely additive if ''a''(''mn'') = ''a''(''m'') + ''a''(''n'') for all natural numbers ''m'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3Blue1Brown
3Blue1Brown is a math YouTube channel created and run by Grant Sanderson. The channel focuses on teaching higher mathematics from a visual perspective, and on the process of discovery and inquiry-based learning in mathematics, which Sanderson calls "inventing math". , the channel has 4.89 million subscribers. Early life and education Sanderson graduated from Stanford University in 2015 with a bachelor's degree in math. He worked for Khan Academy from 2015 to 2016 as part of their content fellowship program, producing videos and articles about multivariable calculus, after which he started focusing his full attention on 3Blue1Brown. Career 3Blue1Brown started as a personal programming project in early 2015. In a podcast of ''Showmakers'', Sanderson explained that he wanted to practice his coding skills and decided to make a graphics library in Python, which eventually became the open-source project "Manim" (short for Mathematical Animation engine). To have a goal for the project ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Hypothesis
In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by , after whom it is named. The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann zeta function ζ(''s'') is a function whose argument ''s'' may be any complex number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime Integers
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Orchard
In mathematics, informally speaking, Euclid's orchard is an array of one-dimensional "trees" of unit height planted at the lattice points in one quadrant of a square lattice. More formally, Euclid's orchard is the set of line segments from to , where and are positive integers. The trees visible from the origin are those at lattice points , where and are coprime, i.e., where the fraction is in reduced form. The name ''Euclid's orchard'' is derived from the Euclidean algorithm. If the orchard is projected relative to the origin onto the plane (or, equivalently, drawn in perspective from a viewpoint at the origin) the tops of the trees form a graph of Thomae's function. The point projects to : \left ( \frac , \frac , \frac \right ). The solution to the Basel problem can be used to show that the proportion of points in the grid that have trees on them is approximately \tfrac and that the error of this approximation goes to zero in the limit Limit or Limits may refe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monatshefte Für Mathematik
'' Monatshefte für Mathematik'' is a peer-reviewed mathematics journal established in 1890. Among its well-known papers is " Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I" by Kurt Gödel, published in 1931. The journal was founded by Gustav von Escherich and Emil Weyr in 1890 as ''Monatshefte für Mathematik und Physik'' and published until 1941. In 1947 it was reestablished by Johann Radon under its current title. It is currently published by Springer in cooperation with the Austrian Mathematical Society. The journal is indexed by ''Mathematical Reviews'' and Zentralblatt MATH. Its 2009 MCQ was 0.58, and its 2009 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a scientometric index calculated by Clarivate that reflects the yearly mean number of citations of articles published in the last two years in a given journal, as i ... was 0.764. External links *''Monatshefte für Mathematik und ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook ''Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime (as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dot Planimeter
A dot planimeter is a device used in planimetrics for estimating the area of a shape, consisting of a transparent sheet containing a square grid of dots. To estimate the area of a shape, the sheet is overlaid on the shape and the dots within the shape are counted. The estimate of area is the number of dots counted multiplied by the area of a single grid square. In some variations, dots that land on or near the boundary of the shape are counted as half of a unit. The dots may also be grouped into larger square groups by lines drawn onto the transparency, allowing groups that are entirely within the shape to be added to the count rather than requiring their dots to be counted one by one. The estimation of area by means of a dot grid has also been called the dot grid method or (particularly when the alignment of the grid with the shape is random) systematic sampling. Perhaps because of its simplicity, it has been repeatedly reinvented. Application In forestry, cartography, and geogr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''a''/''b'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''a''/''b'' and ''α'' may not decrease if ''a''/''b'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of continued fractions. Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as : S^n = \left\ , and an -sphere of radius can be defined as : S^n(r) = \left\ . The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball. In particular: *the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, *a circle, which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
