Athlon 64The Athlon 64 is a ninth-generation, AMD64-architecture microprocessor produced by Advanced Micro Devices (AMD), released on September 23, 2003. It is the third processor to bear the name Athlon, and the immediate successor to the Athlon XP. The second processor (after the Opteron) to implement the AMD64 architecture and the first 64-bit processor targeted at the average consumer, it was AMD's primary consumer CPU, and primarily competed with Intel's Pentium 4, especially the Prescott and Cedar Mill core revisions.
IA-64IA-64 (Intel Itanium architecture) is the instruction set architecture (ISA) of the Itanium family of 64-bit Intel microprocessors. The basic ISA specification originated at Hewlett-Packard (HP), and was subsequently implemented by Intel in collaboration with HP. The first Itanium processor, codenamed Merced, was released in 2001. The Itanium architecture is based on explicit instruction-level parallelism, in which the compiler decides which instructions to execute in parallel.
Windows Server 2003Windows Server 2003, codenamed "Whistler Server", is the second version of Windows Server operating system produced by Microsoft. It is part of the Windows NT family of operating systems and was released to manufacturing on March 28, 2003 and generally available on April 24, 2003. Windows Server 2003 is the successor to the Server editions of Windows 2000 and the predecessor to Windows Server 2008. An updated version, Windows Server 2003 R2, was released to manufacturing on December 6, 2005.
Windows 3.1xWindows 3.1 is a major release of Microsoft Windows. It was released to manufacturing on April 6, 1992, as a successor to Windows 3.0. Like its predecessors, the Windows 3.1 series ran as a shell on top of MS-DOS. Codenamed Janus, Windows 3.1 introduced the TrueType font system as a competitor to Adobe Type Manager. Its multimedia was also expanded, and screensavers were introduced, alongside new software such as Windows Media Player and Sound Recorder. and Control Panel received tweaks, while Windows 3.
Windows PhoneWindows Phone (WP) is a discontinued family of mobile operating systems developed by Microsoft for smartphones as the replacement successor to Windows Mobile and Zune. Windows Phone featured a new user interface derived from the Metro design language. Unlike Windows Mobile, it was primarily aimed at the consumer market rather than the enterprise market. It was first launched in October 2010 with Windows Phone 7.
Euler's criterionIn number theory, Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let p be an odd prime and a be an integer coprime to p. Then Euler's criterion can be concisely reformulated using the Legendre symbol: The criterion first appeared in a 1748 paper by Leonhard Euler. The proof uses the fact that the residue classes modulo a prime number are a field. See the article prime field for more details.
Cubic reciprocityCubic reciprocity is a collection of theorems in elementary and algebraic number theory that state conditions under which the congruence x3 ≡ p (mod q) is solvable; the word "reciprocity" comes from the form of the main theorem, which states that if p and q are primary numbers in the ring of Eisenstein integers, both coprime to 3, the congruence x3 ≡ p (mod q) is solvable if and only if x3 ≡ q (mod p) is solvable. Sometime before 1748 Euler made the first conjectures about the cubic residuacity of small integers, but they were not published until 1849, after his death.
Function (mathematics)In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity).
Analytic continuationIn complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where the infinite series representation which initially defined the function becomes divergent. The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value).
Dirichlet integralIn mathematics, there are several integrals known as the Dirichlet integral, after the German mathematician Peter Gustav Lejeune Dirichlet, one of which is the improper integral of the sinc function over the positive real line: This integral is not absolutely convergent, meaning is not Lebesgue-integrable, because the Dirichlet integral is infinite in the sense of Lebesgue integration. It is, however, finite in the sense of the improper Riemann integral or the generalized Riemann or Henstock–Kurzweil integral.
Zeros and polesIn complex analysis (a branch of mathematics), a pole is a certain type of singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity). Technically, a point z0 is a pole of a function f if it is a zero of the function 1/f and 1/f is holomorphic (i.e. complex differentiable) in some neighbourhood of z0. A function f is meromorphic in an open set U if for every point z of U there is a neighborhood of z in which either f or 1/f is holomorphic.
Complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
