The computation of sine and cosines was similarly done using only integer arithmetic.
Given a positive integer m, A is constant along open intervals m < x < m + 1, and has a jump discontinuity at each integer for which a(m) ≠ 0.
A Transparent Lion: selected poems of Attila József; translated by Michael Castro and Gabor G. Gyukics, Los Angeles, CA: Green Integer 149, 2006 ISBN 1-933382-50-3
An explicit result by Baker and Wüstholz for a linear form Λ with integer coefficients yields a lower bound of the form
In mathematics, in particular in computational algebra, the Berlekamp–Zassenhaus algorithm is an algorithm for factoring polynomials over the integers, named after Elwyn Berlekamp and Hans Zassenhaus.
The constant is irrational; this can be proven with Dirichlet's theorem on arithmetic progressions or Bertrand's postulate (Hardy and Wright, p. 113) or Ramare's theorem that every even integer is a sum of at most six primes.
CTaoCrypt Provides RSA, DSS, Diffie–Hellman, EDH, NTRU, DES, Triple DES, AES (CBC, CTR, CCM, GCM), Camellia, ARC4, HC-128, MD2, MD4, MD5, SHA-1, SHA-2, BLAKE2, RIPEMD-160, Random Number Generation, Large Integer support, and base 16/64 encoding/decoding.
The Degree Confluence Project is a World Wide Web-based all-volunteer project which aims to have people visit each of the integer degree intersections of latitude and longitude on Earth, posting photographs and a narrative of each visit online.
The facility ID number or FIN is a unique positive integer assigned by the United States Federal Communications Commission to each domestic and international (shortwave) broadcast station in its Common Database System (CDBS).
In computer science, Fhourstones is an integer benchmark that efficiently solves positions in the game of Connect-4.
The expression 16 − clz(x − 1)/2 is an effective initial guess for computing the square root of a 32-bit integer using Newton's method.
A Fortunate number, named after Reo Fortune, for a given positive integer n is the smallest integer m > 1 such that pn# + m is a prime number, where the primorial pn# is the product of the first n prime numbers.
Through the use of modulators with frequencies that are non-integer multiples of the carrier signal (i.e. non harmonic), atonal and tonal bell-like and percussive sounds can easily be created.
The problem of extending the factorial to non-integer arguments was apparently first considered by Daniel Bernoulli and Christian Goldbach in the 1720s, and was solved at the end of the same decade by Leonhard Euler.
2005: (Co-translator with Brother Anthony Of Taizé and Kim Young-moo) Ten Thousand Lives by Ko Un, introduction by Robert Hass, (Green Integer: Los Angeles) ISBN 1-933382-06-6
Applesoft BASIC was included in the ROMs of all Apple II models from the Apple II Plus forward, and eventually became the platform for far more programs than Integer BASIC.
Arithmetic function, a term for some functions of an integer variable
Other procedures, especially function-like procedures will have certain behaviours that in specific invocations may enable some work to be avoided: for instance, the Gamma function, if invoked with an integer parameter, could be converted to a calculation involving integer factorials.
This problem, based on a 1770 conjecture by Edward Waring, consists of finding the smallest number g(n) such that every positive integer is the sum of at most g(n) nth powers of positive integers.
Given that the function is analytic within each of these quarters, a nonzero winding number N (always an integer) identifies N zeros of the function inside the quarter in question by Rouché's theorem, each zero counted as many times as its multiplicity.
A model function is determined by interpolating the objective function at (an integer between and ) points; the remaining freedom, if any, is taken up by minimizing the Frobenius norm of the change to the model's Hessian (with respect to the last iteration).
By 1826, he was already using the integer reciprocals of Weiss' coefficients (the intersection of a plane with the three crystallographic axes) to describe the spatial positions of crystal surfaces, from which the British crystallographer William Hallowes Miller (1801-1880) developed the concept of Miller indices in 1839.
A prefix that indicates an integer, e.g., "bin", "tern," or "quatern", may be used in lieu of a numeral, to produce "binary", "ternary", or "quaternary" (2, 3, and 4 states respectively).
#The Encyclopedia of Integer Sequences with Simon Plouffe (1995, ISBN 0-12-558630-2), containing 5,488 sequences and assigned M-numbers from M0000 to M5487.
where denotes the Dirac measure, N is an integer-valued random variable and are random elements of S.
Anatoly Maltsev proved that solvable subgroups of the integer general linear group are polycyclic; and later Louis Auslander (1967) and Swan proved the converse, that any polycyclic group is up to isomorphism a group of integer matrices.
(In another usage ) Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made initial studies of integer Diophantine equations.
Even with grade-school integer multiplication, this is only O((log n)4) time; using the multiplication algorithm with best-known asymptotic running time, the Schönhage–Strassen algorithm, we can lower this to O((log n)3(log log n)(log log log n)) time, or using soft-O notation Õ((log n)3).
Supporting tools: sequential modular simulation, equation based process simulation, AI/expert systems, large-scale nonlinear programming (NLP), optimization of differential algebraic equations (DAEs), mixed-integer nonlinear programming (MINLP), global optimization
HL7 CDA (Clinical Document Architecture) also has a subset of mechanisms similar to (and intended to be compatible with) DICOM for referencing image-related spatial coordinates as observations; it allows for a circle, ellipse, polyline or point to be defined as integer pixel-relative coordinates referencing an external multi-media image object, which may be of a consumer rather than medical image format (e.g., a GIF, PNG or JPEG).
In projective geometry and finite geometry (MSC 51A, 51E, 12K10), a semifield is the analogue of a division algebra, but defined over the integers Z rather than over a field.
Waring's problem asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers.
This conjecture states that there is an effective procedure that, given n ≥ 1 and exponential polynomials in n variables with integer coefficients f1,..., fn, g, produces an integer η ≥ 1 that depends on n, f1,..., fn, g, and such that if α ∈ Rn is a non-singular solution of the system
Divisor function, an arithmetic function giving the number of divisors of an integer.
Draw dual Spectrum type shapes for Amplitudes, Frequencies (with separate: Integer / Decimal parts, and Envelope Variation), Phases - Corresponding to each parameter the base spectrum and the MIDI control spectrum.
The full statement of Vinogradov's theorem gives asymptotic bounds on the number of representations of an odd integer as a sum of three primes.
In 1971 Strassen published another paper together with Arnold Schönhage on asymptotically-fast integer multiplication based on the fast Fourier transform; see the Schönhage–Strassen algorithm.
The solution to Waring's problem for cubes, that every integer is the sum of at most 9 cubes
Ljunggren also posed the question of finding the integer solutions to the Ramanujan–Nagell equation