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Heath-Brown–Moroz Constant
The Heath-Brown–Moroz constant ''C'', named for Roger Heath-Brown and Boris Moroz, is defined as :C=\prod_p\left(1-\frac\right)^7\left(1+\frac\right) = 0.001317641... where ''p'' runs over the primes.Finch, S. R (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. Application This constant is part of an asymptotic estimate for the distribution of rational points of bounded height on the cubic surface In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than a ... ''X''03=''X''1''X''2''X''3. Let ''H'' be a positive real number and ''N''(''H'') the number of solutions to the equation ''X''03=''X''1''X''2''X''3 with all the ''X''''i'' non-negative integers less than or equal to ''H'' and their greatest common divisor equal to 1. Then :N(H)= C \cdot \frac + O(H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roger Heath-Brown
David Rodney "Roger" Heath-Brown FRS (born 12 October 1952), is a British mathematician working in the field of analytic number theory. Education He was an undergraduate and graduate student of Trinity College, Cambridge; his research supervisor was Alan Baker. Career and research In 1979 he moved to the University of Oxford, where from 1999 he held a professorship in pure mathematics. He retired in 2016. Heath-Brown is known for many striking results. He proved that there are infinitely many prime numbers of the form ''x''3 + 2''y''3. In collaboration with S. J. Patterson in 1978 he proved the Kummer conjecture on cubic Gauss sums in its equidistribution form. He has applied Burgess's method on character sums to the ranks of elliptic curves in families. He proved that every non-singular cubic form over the rational numbers in at least ten variables represents 0. Heath-Brown also showed that Linnik's constant is less than or equal to 5.5. More recently, Heath ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boris Moroz
Boris may refer to: People * Boris (given name), a male given name *:''See'': List of people with given name Boris * Boris (surname) * Boris I of Bulgaria (died 907), the first Christian ruler of the First Bulgarian Empire, canonized after his death * Boris II of Bulgaria (c. 931–977), ruler of the First Bulgarian Empire * Boris III of Bulgaria (1894–1943), ruler of the Kingdom of Bulgaria in the first half of the 20th century * Boris, Prince of Tarnovo (born 1997), Spanish-born Bulgarian royal * Boris and Gleb (died 1015), the first saints canonized in Kievan Rus * Boris (singer) (born 1965), pseudonym of French singer Philippe Dhondt Arts and media * Boris (band), a Japanese experimental rock trio * ''Boris'' (EP), by Yezda Urfa, 1975 * "Boris" (song), by the Melvins, 1991 * ''Boris'' (TV series), a 2007–2009 Italian comedy series * '' Boris: The Film'', a 2011 Italian film based on the TV series * '' Boris: The Rise of Boris Johnson'', a 2006 biography by Andrew ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Estimate
In mathematical analysis, asymptotic analysis, also known as asymptotics, is a method of describing limiting behavior. As an illustration, suppose that we are interested in the properties of a function as becomes very large. If , then as becomes very large, the term becomes insignificant compared to . The function is said to be "''asymptotically equivalent'' to , as ". This is often written symbolically as , which is read as " is asymptotic to ". An example of an important asymptotic result is the prime number theorem. Let denote the prime-counting function (which is not directly related to the constant pi), i.e. is the number of prime numbers that are less than or equal to . Then the theorem states that \pi(x)\sim\frac. Asymptotic analysis is commonly used in computer science as part of the analysis of algorithms and is often expressed there in terms of big O notation. Definition Formally, given functions and , we define a binary relation f(x) \sim g(x) \quad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last Theorem may be restated as: for , the Fermat curve of equation x^n+y^n=1 has no other rational points than , , and, if is even, and . Definition Given a field ''k'', and an algebraically closed extension ''K'' of ''k'', an affine variety ''X'' over ''k'' is the set of common zeros in K^n of a collection of polynomials with coefficients in ''k'': :f_1(x_1,\ldots,x_n)=0,\ldots, f_r(x_1,\dots,x_n)=0. These common zeros are called the ''points'' of ''X''. A ''k''-rational point (or ''k''-point) of ''X'' is a point of ''X'' that belongs to ''k''''n'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Height Function
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers. For instance, the ''classical'' or ''naive height'' over the rational numbers is typically defined to be the maximum of the numerators and denominators of the coordinates (e.g. for the coordinates ), but in a logarithmic scale. Significance Height functions allow mathematicians to count objects, such as rational points, that are otherwise infinite in quantity. For instance, the set of rational numbers of naive height (the maximum of the numerator and denominator when expressed in lowest terms) below any given constant is finite despite the set of rational numbers being infinite. In this sense, height functions can be used to prove asymptotic results such as Baker's t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space, and so cubic surfaces are generally considered in projective 3-space \mathbf^3. The theory also becomes more uniform by focusing on surfaces over the complex numbers rather than the real numbers; note that a complex surface has real dimension 4. A simple example is the Fermat cubic surface :x^3+y^3+z^3+w^3=0 in \mathbf^3. Many properties of cubic surfaces hold more generally for del Pezzo surfaces. Rationality of cubic surfaces A central feature of smooth cubic surfaces ''X'' over an algebraically closed field is that they are all rational, as shown by Alfred Clebsch in 1866. That is, there is a one-to-one correspondence defined by rational functions between the projective plane \mathbf^2 minus a lower-dimensional subset and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greatest Common Divisor
In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers. For two integers ''x'', ''y'', the greatest common divisor of ''x'' and ''y'' is denoted \gcd (x,y). For example, the GCD of 8 and 12 is 4, that is, \gcd (8, 12) = 4. In the name "greatest common divisor", the adjective "greatest" may be replaced by "highest", and the word "divisor" may be replaced by "factor", so that other names include highest common factor (hcf), etc. Historically, other names for the same concept have included greatest common measure. This notion can be extended to polynomials (see Polynomial greatest common divisor) and other commutative rings (see below). Overview Definition The ''greatest common divisor'' (GCD) of two nonzero integers and is the greatest positive integer such that is a divisor of both and ; that is, there are integers and such that and , and is the largest s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Constants
A mathematical constant is a key number whose value is fixed by an unambiguous definition, often referred to by a symbol (e.g., an alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory, statistics, and calculus. What it means for a constant to arise "naturally", and what makes a constant "interesting", is ultimately a matter of taste, with some mathematical constants being notable more for historical reasons than for their intrinsic mathematical interest. The more popular constants have been studied throughout the ages and computed to many decimal places. All named mathematical constants are definable numbers, and usually are also computable numbers (Chaitin's constant being a significant exception). Basic mathematical constants These are constants which one is likely to encounter du ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
